# Self-complementary C^{3}-codes and bijective transformations

## Lutz Strüngmann and Elena Fimmel

### Institute for Applied Mathematics

University of Applied Sciences, Mannheim, Germany

Circular codes gained considerable attention as a weaker form of comma-free
codes which were suggested by Crick as a solution for the so-called
frame-shift problem: a sequence of codons can be translated only in a right
frame into the right amino acids. The discovering of an universal code
across species suggested many theoretical and experimental questions.
However, there is a key aspect that relates circular codes to symmetries
and transformations that remains to a large extent unexplored. In this talk
we aim at addressing the issue by studying the symmetries and
transformations that connect different circular codes. The main result is
that the class of 216 C^{3} maximal self-complementary codes can be
partitioned into 27 equivalence classes defined by a particular set of
transformations. We show that such transformations can be put in a group
theoretic framework with an intuitive geometric interpretation. More
general mathematical results about symmetry transformations which are valid
for any kind of circular codes are also presented. Furthermore, we use the
classification in equivalence classes for studying the codon usage of
circular codes in different organisms. The coverage of circular codes
inside equivalence classes allows to characterize the subset of codes
inside the set of 216 that can be obtained from the nucleotide
occurrence in real sequences. Our results pave the way to the study
of the biological consequences of the mathematical structure behind
circular codes and contribute to shed light on the evolutionary steps
that led to the observed symmetries of present codes.

### References

- Elena Fimmel, Simone Giannerini, Diego Luis Gonzalez and Lutz Strüngmann:
Circular codes, symmetries and transformations, submitted for publication
- Gonzalez, D.L., Ginannerini S., Rosa R.: Circular codes revisited:
A statistical approach, Journal of Theoretical Biology, 2011 (275), 1, 21-28
- A.J. Koch, J. Lehmann: About a Symmetry of the genetic code, J. theor. Biol., 1997, 189, 171-174.
- Michel C.J., Pirillo G., Pirillo M.A. (2012). A classification of
20-trinucleotide circular codes. Information and Computation 212, 55-63