On the evolutionary significance of the size and planarity of the proline ring

Proline is a proteinogenic amino acid in which the side chain forms a ring, the pyrrolidine ring. This is a five-membered ring made up of four carbons and one nitrogen. Here, we study the evolutionary significance of this ring size. It is shown that the size of the pyrrolidine ring has the advantage of being nearly planar and strain-free, based on a general mathematical assertion saying that the angular sum of a polygon is maximum if it is planar and convex. We also provide a sketch of the proof to this assertion. The optimality of the ring size of proline can be derived from a triangle inequality for angles. Quasi-planarity is physiologically significant because it allows an easier and evolutionarily old type of fit into binding grooves of proteins with which proline-rich proteins interact. Finally, we present a comparison with other planar, nearly planar and non-planar biomolecules such as neurotransmitters, hormones and toxins, involving, for example, aromatic rings, cyclopentanone and 1,3-dioxole.