Abstract
In order to explore atomic asymmetry and molecular chirality in 2D space, benzenoids composed of 3 to 11 hexagons in 2D space were enumerated in our laboratory. These benzenoids are regarded as planar connected polyhexes and have no internal holes; that is, their internal regions are filled with hexagons. The produced dataset was composed of 357,968 benzenoids, including more than 14 million atoms. Rather than simply labeling the huge number of atoms as being either symmetric or asymmetric, this investigation aims at exploring a quantitative graph theoretical descriptor of atomic asymmetry. Based on the particular characteristics in the 2D plane, we suggested the weighted atomic sum as the descriptor of atomic asymmetry. This descriptor is measured by circulating around the molecule going in opposite directions. The investigation demonstrates that the weighted atomic sums are superior to the previously reported quantitative descriptor, atomic sums. The investigation of quantitative descriptors also reveals that the most asymmetric atom is in a structure with a spiral ring with the convex shape going in clockwise direction and concave shape going in anticlockwise direction from the atom. Based on weighted atomic sums, a weighted F index is introduced to quantitatively represent molecular chirality in the plane, rather than merely regarding benzenoids as being either chiral or achiral. By validating with enumerated benzenoids, the results indicate that the weighted F indexes were in accordance with their chiral classification (achiral or chiral) over the whole benzenoids dataset. Furthermore, weighted F indexes were superior to previously available descriptors. Benzenoids possess a variety of shapes and can be extended to practically represent any shape in 2D space—our proposed descriptor has thus the potential to be a general method to represent 2D molecular chirality based on the difference between clockwise and anticlockwise sums around a molecule.
Highlights
Many molecular properties are dependent on the molecular shape
The results showed that the atomic sums of 275,022 asymmetric atoms were zeros, i.e., 1.9% of asymmetric atoms (
The weighted atomic sums were obtained by circulating around the molecule in opposite directions based on the features in 2D space, and the weighted F indexes were derived from weighted atomic sums
Summary
Many molecular properties are dependent on the molecular shape. When some proteins are active, these protein molecules may double over or twist into radically different shapes. Randić approached a descriptor of 2D molecular chirality of benzenoids, called F index, which was derived from atomic asymmetry. Our studies on the degree of chirality of benzenoids tried to exploit a practical method to represent 2D chirality based on the pioneering work of Randić. This investigation aimed at exploring the improved descriptors, in their ability to overcome the observed limitations of the Randić’s descriptors, and in their ability to represent the atomic asymmetry from the symmetrical atoms to the most asymmetric atoms, as well as in their ability to represent the molecular chirality from the achiral molecules to the most chiral molecules. The shapes of benzenoids can be regarded as shapes of graphite and graphene, and have potential to approximately represent similar shapes [19]
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