The big kinship of proline: Counting saturated and unsaturated secondary amino acids of different ring size

Proline is a well-known proteinogenic amino acid, which features a unique structure among the canonical amino acids: its side chain forms a ring with the backbone nitrogen atom. There are many other cyclic secondary amino acids of different ring size, which are non-proteinogenic. Many of their derivatives are natural products and have high potential in pharmacology, biotechnology, and synthetic biology. Mathematical and computational approaches in biochemistry complement the analysis of the physicochemical and functional properties of compounds, and provide an overview of both known and hitherto unknown, yet hypothetically possible, molecules. Here, we present a mathematical enumeration procedure for all possible cyclic secondary α-amino and imino acids, depending on the number of atoms in the ring, in terms of saturated and unsaturated congeners. A recursion formula is given, which results in a modified Lucas sequence. Moreover, explicit formulas for determining the number of such substances are derived. This enumeration is helpful for the construction of virtual compound databases and computer-assisted design of chemical syntheses. Additionally, we provide an overview of detected and characterised structures in this class, as well as some of their derivatives, along with exemplary applications. Moreover, we discuss the conformational and functional properties of compounds with different ring sizes.