Theory in Biosciences

, Volume 137, Issue 1, pp 51–65 | Cite as

Self-complementary circular codes in coding theory

  • Elena Fimmel
  • Christian J. Michel
  • Martin Starman
  • Lutz Strüngmann
Original Article


Self-complementary circular codes are involved in pairing genetic processes. A maximal \(C^3\) self-complementary circular code X of trinucleotides was identified in genes of bacteria, archaea, eukaryotes, plasmids and viruses (Michel in Life 7(20):1–16 2017, J Theor Biol 380:156–177, 2015; Arquès and Michel in J Theor Biol 182:45–58 1996). In this paper, self-complementary circular codes are investigated using the graph theory approach recently formulated in Fimmel et al. (Philos Trans R Soc A 374:20150058, 2016). A directed graph \(\mathcal {G}(X)\) associated with any code X mirrors the properties of the code. In the present paper, we demonstrate a necessary condition for the self-complementarity of an arbitrary code X in terms of the graph theory. The same condition has been proven to be sufficient for codes which are circular and of large size \(\mid X \mid \ge 18\) trinucleotides, in particular for maximal circular codes (\(\mid X \mid = 20\) trinucleotides). For codes of small-size \(\mid X \mid \le 16\) trinucleotides, some very rare counterexamples have been constructed. Furthermore, the length and the structure of the longest paths in the graphs associated with the self-complementary circular codes are investigated. It has been proven that the longest paths in such graphs determine the reading frame for the self-complementary circular codes. By applying this result, the reading frame in any arbitrary sequence of trinucleotides is retrieved after at most 15 nucleotides, i.e., 5 consecutive trinucleotides, from the circular code X identified in genes. Thus, an X motif of a length of at least 15 nucleotides in an arbitrary sequence of trinucleotides (not necessarily all of them belonging to X) uniquely defines the reading (correct) frame, an important criterion for analyzing the X motifs in genes in the future.


Self-complementary circular codes Graph properties Translation process Reading frame Genetic code 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Elena Fimmel
    • 1
  • Christian J. Michel
    • 2
  • Martin Starman
    • 1
  • Lutz Strüngmann
    • 1
  1. 1.Faculty for Computer Sciences, Institute of Mathematical BiologyMannheim University of Applied SciencesMannheimGermany
  2. 2.Theoretical Bioinformatics, ICube, CNRSUniversity of StrasbourgIllkirchFrance

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