Combinatorics of aliphatic amino acids

This study combines biology and mathematics, showing that a relatively simple question from molecular biology can lead to complicated mathematics. The question is how to calculate the number of theoretically possible aliphatic amino acids as a function of the number of carbon atoms in the side chain. The presented calculation is based on earlier results from theoretical chemistry concerning alkyl compounds. Mathematical properties of this number series are highlighted. We discuss which of the theoretically possible structures really occur in living organisms, such as leucine and isoleucine with a chain length of four. This is done both for a strict definition of aliphatic amino acids only involving carbon and hydrogen atoms in their side chain and for a less strict definition allowing sulphur, nitrogen and oxygen atoms. While the main focus is on proteinogenic amino acids, we also give several examples of non-proteinogenic aliphatic amino acids, playing a role, for instance, in signalling. The results are in agreement with a general phenomenon found in biology: Usually, only a small number of molecules are chosen as building blocks to assemble an inconceivable number of different macromolecules as proteins. Thus, natural biological complexity arises from the multifarious combination of building blocks.