New mathematical measures for apprehending complexity of chiral molecules using information entropy

In this paper, we present several new theoretical measures based on information entropy that can be used to analyze the information content of a chiral molecule. Starting from a differentiation between "chiral" and "achiral" portions in a chiral molecule, we define a new concept that allows us to quantify the complexity of chiral constitutional 2D-isomers of C₁₀ to C₂₀ alkanes. Various new chiral and achiral information measures founded on joint entropy, mutual information, and conditional entropy are presented providing an access to a set of regression equations. Then, introducing a case-based measure of entropy, we demonstrate that the distribution of the chiral complexity in these molecules is mostly skewed-right: 60% of the chiral isomers follow a 60/40 distribution rule, which indicates a concentration of chiral complexity in a small number of topological features. Furthermore, by replacing 2D topological distances by 3D distances, the application of these new information measures goes from conformational to racemization and deracemization studies. Interestingly, when the geometrical distances between atoms and the chiral center(s) are taken into account when determining the chiral information entropy, one can observe a significative Pearson correlation coefficient (R = 0.70) between the chiral entropy of 3D molecules and the continuous chirality measure. Finally, we show that our approach is applicable to almost any type of chiral organic chemical structures if in the entropy equation, atoms are represented by their electrotopological state (E-state) index instead of connectivity.

Enumerating Tree-Like Graphs and Polymer Topologies with a Given Cycle Rank

Cycle rank is an important notion that is widely used to classify, understand, and discover new chemical compounds. We propose a method to enumerate all non-isomorphic tree-like graphs of a given cycle rank with self-loops and no multiple edges. To achieve this, we develop an algorithm to enumerate all non-isomorphic rooted graphs with the required constraints. The idea of our method is to define a canonical representation of rooted graphs and enumerate all non-isomorphic graphs by generating the canonical representation of rooted graphs. An important feature of our method is that for an integer n≥1, it generates all required graphs with n vertices in O(n) time per graph and O(n) space in total, without generating invalid intermediate structures. We performed some experiments to enumerate graphs with a given cycle rank from which it is evident that our method is efficient. As an application of our method, we can generate tree-like polymer topologies of a given cycle rank with self-loops and no multiple edges.