# On Dichotomic Classes and Bijections of the Genetic Code

## Elena Fimmel and Lutz Strüngmann

### Institute for Applied Mathematics

University of Applied Sciences, Mannheim, Germany

In the talk decompositions of the set of 64 codons in two equal sized
subsets, called dichotomic classes, are presented. Dichotomic classes allow to
uncover many symmetry properties of the code and mirror biochemical features
occurring at discrete base positions in the codon.

A recent mathematical model of the genetic code (see, for instance, [1],
[2], [3], [4]), based on non-power representation of integer numbers,
provides an original explanation for the structural properties and the
degeneracy distribution that naturally occur in the genetic code. In
particular, a known biologically meaningful Rumer’s dichotomy (see, for
instance, [5]) and a new chemically meaningful parity dichotomy arise
based on the model in a natural way. In our work we show that the
codon/anticodon complementarity can be understood and obtained in the same
algorithmic way as the Rumer’s and the parity dichotomies and present a
general algorithm underlying significant dichotomic partitions of the code.

In addition the global framework of bijective transformations of the
nucleotide bases is also discussed and we clarify when dichotomic partitions
can be generated. Interestingly, the algorithm underlying dichotomic class
definition mirrors operations that have a factual biological function in the
decoding center of the ribosome.

### References

- Fimmel, E., Danielli A., Strüngmann L.: On dichotomic classes and
bijections of the genetic code. J. Theor. Biol. (2013),
http://dx.doi.org/10.1016/j.jtbi.2013.07.027
- S. Giannerini, D.L. Gonzalez, R. Rosa: DNA, dichotomic classes and frame
syncronization: a quasi-crystal framework., Phil. Trans. R. Soc., 2012, 370,
pp. 2987-3006
- D.L. Gonzalez, S. Giannerini, R. Rosa: Strong short-range correlations and
dichotomic codon classes in coding DNA sequences, Phys Rev E Stat Nonlin Soft
Matter Phys. 2008 Nov;78(5 Pt 1):051918. Epub 2008 Nov 19.
- D.L. Gonzalez: The Mathematical Structure of the Genetic Code, In
Barbieri (Ed.): Codes of Life: The Rules of Microevolution, Springer, 2008
- Y.B. Rumer: Systematization of codons in the genetic code, Doklady
Akademii nauk SSSR 1969 Aug;187(4):937-8.