Enumeration of saturated and unsaturated substituted N-heterocycles

TL;DR

This paper develops mathematical methods to enumerate all possible saturated and unsaturated nitrogen-containing heterocycles with substituents, aiding virtual database construction and synthesis planning.

Contribution

It introduces a recursive enumeration procedure, explicit formulas involving the Golden Ratio and binomial coefficients, for nitrogen heterocycles with substituents, expanding computational chemistry tools.

Findings

Derived a recursion formula linked to Lucas numbers

Provided explicit formulas based on the Golden Ratio and binomial coefficients

Facilitates virtual compound database expansion and synthesis planning

Abstract

Mathematical and computational approaches in chemistry and biochemistry fill a gap in respect to the analysis of the physicochemical features of compounds and their functionality and provide an overview of known as well as yet unknown, but hypothetically possible structures. Nitrogen-containing heterocycles such as aziridine, azetidine and pyrrolidine bear a high potential in pharmacology, biotechnology and synthetic biology. Here, we present a mathematical enumeration procedure for all possible azaheterocycles with at least one substituent depending on the number of atoms in the ring, in the sense of saturated and unsaturated congeners. One subgroup belonging to that substance class is constituted by ring-shaped amino acids with a secondary amino group, such as proline. A recursion formula is derived, which results in a modified Lucas number series. Moreover, an explicit formula for determining the number of such substances based on the Golden Ratio is given and a second one, based on binomial coefficients, is newly derived. This enumeration is a helpful tool for construction or complementation of virtual compound databases and for computer-assisted chemical synthesis route planning.