aliphatic cluster, fluorine and chlorine from the halogen cluster, N-acetyl and phenol from the substituents showing hydrophilicity, and a range of electronic and bulk values. Including hydrogen, there will be 73, or 343, different combinations. Obviously, that is too many for an initial evaluation. Instead, certain rules have been devised to maximize the information obtained from a minimum number of compounds. These include the following:

1. Each substituent must occur more than once at each position on which it is found.

2. The number of times that each substituent at a particular position appears should be approximately equal.

3. No two substituents should be present in a constant combination.

4. When combinations of substituents are a necessity, they should not occur more frequently than any other combination.

Following these guidelines, the initial test set can be reduced to 24 to 26 compounds. Depending on the precision of the biological tests, it will be possible to see if the data will fit a QSAR model. Even an approximate model usually will indicate the types of substituents to test further and what positions on the molecules are sensitive to substitution and, if sensitive, to what degree variation in lipophilic, electronic, or bulk character is important. Just to ensure that the model is valid, it is a good idea to synthesize a couple of compounds that the model predicts would be inactive. As each group of new compounds is tested, the QSAR model is refined until the investigators have a pretty good idea what substituent patterns are important for the desired activity. These same techniques used to develop potent compounds with desired activity also can be used to evaluate the influence of substituent patterns on undesired toxic effects and pharmacokinetic properties.

Topological Descriptors

An alternate method of describing molecular structure is based on graph theory, in which the bonds connecting the atoms is considered a path that is traversed from one atom to another. Consider Figure 2.12 containing d-phenylalanine and its hydrogen-suppressed graph representation. The numbering is arbitrary and not based on International Union of Pure and Applied Chemistry (IUPAC) or Chemical Abstracts nomenclature rules. A connectivity table, Table 2.7, is constructed.

Table 2.7 is a two-dimensional connectivity table for the hydrogen-suppressed phenylalanine molecule. No 3D representation is implied. Further, this type of connectivity table will be the same for molecules with asymmetric atoms (D vs. L) or for those that can exist in more than one conformation (i.e., chair vs. boat conformation, anti vs. gauche vs. eclipsed).

Graph theory is not limited to the paths followed by chemical bonds. In its purest form, the atoms in the phenyl ring of phenylalanine would have paths connecting atom 7 with atoms 9, 10, 11, and 12; atom 8 with atoms 10, 11, and 12; atom 9 with atoms 11 and 12; and atom 10 with atom 12. Also, the graph itself might differentiate neither single, double, and triple bonds nor the type of atom (C, O, and N in the phenylalanine example). Connectivity tables can be coded to indicate the type of bond.

The most common application of graph theory used by medicinal chemistry is called molecular connectivity. It limits the paths to the molecule's actual chemical bonds. Table 2.8 shows several possible paths for phenylalanine, including linear paths and clusters or branching. Numerical values for each path or path-cluster are based on the number of nonhydrogen bonds to each atom. Let us examine oxygen atom 1. There is only one nonhydrogen bond, and it connects oxygen atom 1 to carbon atom 2. The formula is the reciprocal square root of the number of bonds. For oxygen 1, the connectivity value is 1. For carbonyl oxygen 2, it is 2-1/2, or 0.707. Note that there is no difference between oxygen 1 and nitrogen 5. Both have only one nonhydrogen bond and a connectivity value of 1. Similarly, there is no difference in values for a carbonyl oxygen and a methylene carbon, each having two nonhydrogen bonds. The final connectivity values for a path are the reciprocal square roots of the products of each path. For the second-order path 2C-4C-6C, the reciprocal square root (3 X 3 X 2)-1/2 is 4.243. The values for each path order are calculated and summed.

As noted previously, the method as described so far cannot distinguish between atoms that have the same number of nonhydrogen bonds. A method to distinguish heteroatoms from each other and carbon atoms is based on the difference between the number of valence electrons and possible hydrogen atoms (which are suppressed in the graph). The valence connectivity term for an alcoholic oxygen would be 6 valence electrons minus 1 hydrogen, or 5. The valence connectivity term for a primary amine

Figure 2.12 • Hydrogen-suppressed graphic representation of phenylalanine.

TABLE 2.8 Examples of Paths Found in the Phenylalanine Molecule

1st Order Path 2nd Order Path 3rd Order Path 4th Order Path 5th Order Path Path-Cluster

TABLE 2.8 Examples of Paths Found in the Phenylalanine Molecule

1st Order Path 2nd Order Path 3rd Order Path 4th Order Path 5th Order Path Path-Cluster

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