Enumerating stereo-isomers of tree-like polyinositols

Enumeration of molecules is one of the fundamental problems in bioinformatics and chemoinformatics which is also important from a practical viewpoint. We consider the problem of enumerating the stereo-isomers of tree-like polyinositol molecules (with chemical formula $\hbox {C}_{6n}\hbox {O}_{5n+6}\hbox {H}_{4n+2}$ where $n$ is the number of hexagonal oinositol rings) and monosubstituted tree-like polyinositols (with chemical formula $\hbox {C}_{6n}\hbox {O}_{5n+6}\hbox {H}_{4n+1}\hbox {Z}$ ). We establish recursion counting formulas for the numbers of the stereo-isomers for these two classes of molecules, in which chirality is also taken into account. In our study, the generating function, Pólya enumeration theory and ‘Dissimilarity Characteristic Theorem’ play important roles. Compared to some known computer programs such as ISOMERS, MOLGEN, exhaustive construction and Dynamic Programming etc., our method is more efficient to our enumeration problem with larger number of inositol rings. Further more, based on the obtained recursion formulas, we derive the asymptotic values for the numbers of these two stereo-isomers from which we conclude that almost all tree-like and monosubstituted tree-like polyinositols are chiral.