Sphericities of Double Cosets,Double Coset Representations,and Fujita’s Proligand Method for Combinatorial Enumeration of Stereoisomers

The concepts of double coset representations and sphericities of double cosets are proposed to characterize stereoisomerism, where double cosets are classified into three types, i.e., homospheric double cosets, enantiospheric double cosets, or hemispheric double cosets. They determine modes of substitutions (i.e., chirality fittingness), where homospheric double cosets permit achiral ligands only; enantiospheric ones permit achiral ligands or enantiomeric pairs; and hemispheric ones permit achiral and chiral ligands. The sphericities of double cosets are linked to the sphericities of cycles which are ascribed to right coset representations. Thus, each cycle is assigned to the corresponding sphericity index (a _d , c _d , or b _d ) so as to construct a cycle indices with chirality fittingness (CI-CFs). The resulting CI-CFs are proved to be identical with CI-CFs introduced in Fujita’s proligand method (S. Fujita, Theor. Chem. Acc. 113 (2005) 73–79 and 80–86). The versatility of the CI-CFs in combinatorial enumeration of stereoisomers is demonstrated by using methane derivatives as examples, where the numbers of achiral plus chiral stereoisomers, those of achiral stereoisomers, and those of chiral stereoisomers are calculated separately by means of respective generating functions.     

Doubly-Colored Graphs as Graphical Models for Subductions of Coset Representations, Double Cosets, and Unit Subduced Cycle Indices

The concept of doubly-colored graphs is proposed to model subductions of coset representations, double cosets, and unit subduced cycle indices, which have been mathematically formulated in coset algebraic theory developed by Fujita (Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag, Berlin, 1991).     

The Thiophene “Sigma‐Hole” as a Concept for Preorganized,Specific Recognition of G⋅C Base Pairs in the DNA Minor Groove

In spite of its importance in cell function, targeting DNA is under‐represented in the design of small molecules. A barrier to progress in this area is the lack of a variety of modules that recognize G ? C base pairs (bp) in DNA sequences. To overcome this barrier, an entirely new design concept for modules that can bind to mixed G ? C and A ? T sequences of DNA is reported herein. Because of their successes in biological applications, minor‐groove‐binding heterocyclic cations were selected as the platform for design. Binding to A ? T sequences requires hydrogen‐bond donors whereas recognition of the G‐NH₂ requires an acceptor. The concept that we report herein uses pre‐organized N‐methylbenzimidazole (N‐MeBI) thiophene modules for selective binding with mixed bp DNA sequences. The interaction between the thiophene sigma hole (positive electrostatic potential) and the electron‐donor nitrogen of N‐MeBI preorganizes the conformation for accepting an hydrogen bond from G‐NH₂. The compound–DNA interactions were evaluated with a powerful array of biophysical methods and the results show that N‐MeBI‐thiophene monomer compounds can strongly and selectively recognize single G ? C bp sequences. Replacing the thiophene with other moieties significantly reduces binding affinity and specificity, as predicted by the design concept. These results show that the use of molecular features, such as sigma‐holes, can lead to new approaches for small molecules in biomolecular interactions.