, Volume 98, Issue 1, pp 79–86 | Cite as

Combinatorics of aliphatic amino acids

  • Konrad GrützmannEmail author
  • Sebastian Böcker
  • Stefan Schuster
Original Paper


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.


Aliphatic amino acids Aliphatic side chain Amino acid signalling Enumeration of isomers Pólya’s enumeration theorem Ternary tree graphs 



We kindly thank Gunnar Brinkmann from the University of Gent, Belgium, for suggestions about the very first ideas for this manuscript. Further acknowledgements go to Dr Ina Weiß and Heike Göbel for literature search. We also thank Christian Bodenstein who inspired us to the idea of plotting the carbon ‘investment’ (see Fig. 4).

Conflict of interests

The authors declare that they have no conflict of interest.


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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Konrad Grützmann
    • 1
    Email author
  • Sebastian Böcker
    • 1
  • Stefan Schuster
    • 1
  1. 1.Jena Centre for BioinformaticsFriedrich Schiller University JenaJenaGermany

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