Chirality measures

One example of a quantitative measure of molecular chirality is the continuous chirality measure (CCM) [39, 40]. It was developed in the broader context of continuous symmetry measures. A chital object can be defined as an object that lacks improper elements of symmetry (mirror plane, center of inversion, or improper rotation axes). The farther it is from a situation in which it would have an improper element of symmetry, the higher its continuous chirality measure. 

Continuous chirality measure is then defined as follows given a configuration of points P = I, its chirality content is determined by finding the nearest configuration of points Pi - 2 which has an improper element of symmetry, and by calculating the distance between the two sets using Eq. (26). 

The S(Gj ) thus obtained is the minimal chirality measure of the given configuration, on a continuous scale of 0 S 100. 

Application of the CCM to small sets (n < 6) of enzyme inhibitors revealed correlations between the inhibitory activity and the chirality measure of the inhibitors, calculated by Eq. (26) for the entire structure or for the substructure that interacts with the enzyme (pharmacophore) [41], This was done for arylammonium inhibitors of trypsin, Di-dopamine receptor inhibitors, and organophosphate inhibitors of trypsin, acetylcholine esterase, and butyrylcholine esterase. Because the CCM values are equal for opposite enantiomers, the method had to be applied separately to the two families of enantiomers (R- and S-enantiomers). 

For our purposes, a well-suited chirality measure must have four features ... 

Our chirality measure is defined in terms of the principal axes coordinates by... 

Mezey, P. G. (1997) Chirality measures and graph representations. Computers Math. Applic. 34, 105-112. 

Mezey, P. G. (1998) The proof of the metric properties of a fuzzy chirality measure of molecular electron density clouds. J. Molec. Struct. (Theochem.) 455, 183-190. 

How can chirality be quantified There are two minimum requirements. First, recall that an object X (no matter whether physical or mathematically abstract) is chiral if and only if it is nonsuperposable on its mirror image X (X X). It follows that any chirality measure % that quantifies this property can equal zero if and only if the object is achiral any function that does not satisfy this conditio sine qua non fails to qualify as a measure of chirality147-148 ... 

It has been recognized147 that chirality measures can be subdivided into two types those that gauge the extent to which a chiroid differs from an achiral reference object (measures of the first kind) and those that gauge the extent to which two enantiomorphs differ from one another (measures of the second kind). In chirality measures of the first kind, the question to be answered is How dissimilar are the chiroid and its achiral reference object In chirality measures of the second kind, the question is How dissimilar are the two enantiomorphs of a chiroid In both cases the underlying concept is that of a distance, measured either between a chiral and an achiral object or between two enantiomorphous chiroids. That is, the degree of chirality of a chiroid X is defined in relation to another, chiral or achiral, reference object Xref The less these two objects match, the more chiral is X. 

Consider, for example, the question What is the shape of the most chiral right triangle in R2 The answer to this question depends on the particular function that is chosen as the chirality measure According to measures based on geometric chirality products,151 on symmetry coordinates,152 on common volumes,153 and on Hausdorff distances,131 154 the smallest internal angles are 18.8°, 30.0°, 37.5°, and 35.2°, respectively.147 The function x(0) = sin 40 mentioned earlier achieves its maximum [ (0) = 1] at 0 = 22.5°. Because there are as many answers to this question as there are functions that can measure the chirality of a triangle, and because there is, in principle, no limit to the number of such functions, the question remains, in a deep sense, unanswerable. 

In stark contrast to the numerous functions that are available to measure geometrical chirality, no measure has yet been reported for the quantification of topological chirality. In analogy to geometrical chirality measures, topological chirality measures %(K) must satisfy two minimal conditions They can be equal to zero if and only if the knot or link is achiral, and they have to have the same absolute value for two topological enantiomorphs. 

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