Lutz Strüngmann

Circular codes, symmetries and transformations

Circular codes, putative remnants of primeval comma-free codes, have gained considerable attention in the last years. In fact they represent a second kind of genetic code potentially involved in detecting and maintaining the normal reading frame in protein coding sequences. 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 article 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

On dichotomic classes and bijections of the genetic code.

C^3

On the hierarchy of trinucleotide n-circular codes and their corresponding amino acids

C3 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. 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.

On models of the genetic code generated by binary dichotomic algorithms

The circular code theory proposes that genes are constituted of two trinucleotide codes: the classical genetic code with 61 trinucleotides for coding the 20 amino acids (except the three stop codons {TAA,TAG,TGA}) and a circular code based on 20 trinucleotides for retrieving, maintaining and synchronizing the reading frame. It relies on two main results: the identification of a maximal C3 self-complementary trinucleotide circular code X in genes of bacteria, eukaryotes, plasmids and viruses (Michel 2015 J. Theor. Biol. 380, 156–177. (doi:10.1016/j.jtbi.2015.04.009); Arquès & Michel 1996 J. Theor. Biol. 182, 45–58. (doi:10.1006/jtbi.1996.0142)) and the finding of X circular code motifs in tRNAs and rRNAs, in particular in the ribosome decoding centre (Michel 2012 Comput. Biol. Chem. 37, 24–37. (doi:10.1016/j.compbiolchem.2011.10.002); El Soufi & Michel 2014 Comput. Biol. Chem. 52, 9–17. (doi:10.1016/j.compbiolchem.2014.08.001)). The univerally conserved nucleotides A1492 and A1493 and the conserved nucleotide G530 are included in X circular code motifs. Recently, dinucleotide circular codes were also investigated (Michel & Pirillo 2013 ISRN Biomath. 2013, 538631. (doi:10.1155/2013/538631); Fimmel et al. 2015 J. Theor. Biol. 386, 159–165. (doi:10.1016/j.jtbi.2015.08.034)). As the genetic motifs of different lengths are ubiquitous in genes and genomes, we introduce a new approach based on graph theory to study in full generality n-nucleotide circular codes X, i.e. of length 2 (dinucleotide), 3 (trinucleotide), 4 (tetranucleotide), etc. Indeed, we prove that an n-nucleotide code X is circular if and only if the corresponding graph is acyclic. Moreover, the maximal length of a path in corresponds to the window of nucleotides in a sequence for detecting the correct reading frame. Finally, the graph theory of tournaments is applied to the study of dinucleotide circular codes. It has full equivalence between the combinatorics theory (Michel & Pirillo 2013 ISRN Biomath. 2013, 538631. (doi:10.1155/2013/538631)) and the group theory (Fimmel et al. 2015 J. Theor. Biol. 386, 159–165. (doi:10.1016/j.jtbi.2015.08.034)) of dinucleotide circular codes while its mathematical approach is simpler.

BREAKING UP FINITE AUTOMATA PRESENTABLE TORSION-FREE ABELIAN GROUPS

Dichotomic classes arising from a recent mathematical model of the genetic code allow to uncover many symmetry properties of the code, and although theoretically derived, they permitted to build statistical classifiers able to retrieve the correct translational frame of coding sequences. Herein we formalize the mathematical properties of these classes, first focusing on all the possible decompositions of the 64 codons of the genetic code into two equally sized dichotomic subsets. Then the global framework of bijective transformations of the nucleotide bases is discussed and we clarify when dichotomic partitions can be generated. In addition, we show that the parity dichotomic classes of the mathematical model and complementarity dichotomic classes obtained in the present article can be formalized in the same algorithmic way the dichotomic Rumers degeneracy classes. Interestingly, we find that the algorithm underlying dichotomic class definition mirrors biochemical features occurring at discrete base positions in the decoding center of the ribosome.

Maximal dinucleotide comma-free codes.

The presence of circular codes in mRNA coding sequences is postulated to be involved in informational mechanisms aimed at detecting and maintaining the normal reading frame during protein synthesis. Most of the recent research is focused on trinucleotide circular codes. However, also dinucleotide circular codes are important since dinucleotides are ubiquitous in genomes and associated to important biological functions. In this work we adopt the group theoretic approach used for trinucleotide codes in Fimmel et al. (2015) to study dinucleotide circular codes and highlight their symmetry properties. Moreover, we characterize such codes in terms of n-circularity and provide a graph representation that allows to visualize them geometrically. The results establish a theoretical framework for the study of the biological implications of dinucleotide circular codes in genomic sequences.

Yury Borisovich Rumer and his ‘biological papers’ on the genetic code

Circular codes are putative remnants of primeval comma-free codes and are potentially involved in detecting and maintaining the normal reading frame in protein coding sequences. In Michel and Pirillo (2013a) it was shown by computer algorithm that no maximal trinucleotide circular code can encode more than 18 different amino acids under the standard version of the genetic code. For comma-free codes the maximum is even less, namely 13 (Michel, 2014). The main purpose of this paper is to investigate these facts from a mathematical point of view and to show why the codes with the best-known error detecting properties are limited in the number of amino acids they can encode. We introduce five hierarchically ordered classes of trinucleotide codes including the well-known comma-free and circular codes and prove combinatorically that it is impossible to encode all amino acids using codes from four out of the five classes that have the strongest error detecting properties. However, it is possible to encode all 20 amino acids using codes from the largest class with the weakest properties. Additionally, we develop a handy criterion for circularity. As an application, it is shown that all codes from a special class of trinucleotide codes which includes the RNY-primeval code (Shepherd, 1986) are automatically circular. We also list which amino acids these codes encode.

Codon Distribution in Error-Detecting Circular Codes.

In this paper we introduce the concept of a BDA-generated model of the genetic code which is based on binary dichotomic algorithms (BDAs). A BDA-generated model is based on binary dichotomic algorithms (BDAs). Such a BDA partitions the set of 64 codons into two disjoint classes of size 32 each and provides a generalization of known partitions like the Rumer dichotomy. We investigate what partitions can be generated when a set of different BDAs is applied sequentially to the set of codons. The search revealed that these models are able to generate code tables with very different numbers of classes ranging from 2 to 64. We have analyzed whether there are models that map the codons to their amino acids. A perfect matching is not possible. However, we present models that describe the standard genetic code with only few errors. There are also models that map all 64 codons uniquely to 64 classes showing that BDAs can be used to identify codons precisely. This could serve as a basis for further mathematical analysis using coding theory, for example. The hypothesis that BDAs might reflect a molecular mechanism taking place in the decoding center of the ribosome is discussed. The scan demonstrated that binary dichotomic partitions are able to model different aspects of the genetic code very well. The search was performed with our tool Beady-A. This software is freely available at http://mi.informatik.hs-mannheim.de/beady-a. It requires a JVM version 6 or higher.