Circular Tessera Codes in the Evolution of the Genetic Code

Circular mixed sets

In this article, we introduce the new mathematical concept of circular mixed sets of words over an arbitrary finite alphabet. These circular mixed sets may not be codes in the classical sense and hence allow a higher amount of information to be encoded. After describing their basic properties, we generalize a recent graph theoretical approach for circularity and apply it to distinguish codes from sets (i.e. non-codes). Moreover, several methods are given to construct circular mixed sets. Finally, this approach allows us to propose a new evolution model of the present genetic code that could have evolved from a dinucleotide world to a trinucleotide world via circular mixed sets of dinucleotides and trinucleotides.

Robustness against point mutations of genetic code extensions under consideration of wobble-like effects

Many theories of the evolution of the genetic code assume that the genetic code has always evolved in the direction of increasing the supply of amino acids to be encoded (Barbieri, 2019; Di Giulio, 2005; Wong, 1975). In order to reduce the risk of the formation of a non-functional protein due to point mutations, nature is said to have built in control mechanisms. Using graph theory the authors have investigated in Blazej et al. (2019) if this robustness is optimal in the sense that a different codon-amino acid assignment would not generate a code that is even more robust. At present, efforts to expand the genetic code are very relevant in biotechnological applications, for example, for the synthesis of new drugs (Anderson et al., 2004; Chin, 2017; Dien et al., 2018; Kimoto et al., 2009; Neumann et al., 2010). In this paper we generalize the approach proposed in Blazej et al. (2019) and will explore hypothetical extensions of the standard genetic code with respect to their optimal robustness in two ways: (1) We keep the usual genetic alphabet but move from codons to longer words, such as tetranucleotides. This increases the supply of coding words and thus makes it possible to encode non-canonical amino acids. (2) We expand the genetic alphabet by introducing non-canonical base pairs. In addition, the approach from Blazej et al. (2019) and Blazej et al. (2018) is extended by incorporating the weights of single point-mutations into the model. The weights can be interpreted as probabilities (appropriately normalized) or degrees of severity of a single point mutation. In particular, this new approach allows us to take a closer look at the wobble effects in the translation of codons into amino acids. According to the results from Blazej et al. (2019) and Blazej et al. (2018), the standard genetic code is not optimal in terms of its robustness to point mutations if the weights of single point mutations are not taken into account. After incorporation into the model weights that mimic the wobble effect, the results of the present work show that it is much more robust, almost optimal in that respect. We hope, that this theoretical analysis might help to assess extended genetic codes and their abilities to encode new amino acids.

Equivalence classes of circular codes induced by permutation groups

In the 1950s, Crick proposed the concept of so-called comma-free codes as an answer to the frame-shift problem that biologists have encountered when studying the process of translating a sequence of nucleotide bases into a protein. A little later it turned out that this proposal unfortunately does not correspond to biological reality. However, in the mid-90s, a weaker version of comma-free codes, so-called circular codes, was discovered in nature in J Theor Biol 182:45–58, 1996. Circular codes allow to retrieve the reading frame during the translational process in the ribosome and surprisingly the circular code discovered in nature is even circular in all three possible reading-frames (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^3$$\end{document}C3-property). Moreover, it is maximal in the sense that it contains 20 codons and is self-complementary which means that it consists of pairs of codons and corresponding anticodons. In further investigations, it was found that there are exactly 216 codes that have the same strong properties as the originally found code from J Theor Biol 182:45–58. Using an algebraic approach, it was shown in J Math Biol, 2004 that the class of 216 maximal self-complementary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^3$$\end{document}C3-codes can be partitioned into 27 equally sized equivalence classes by the action of a transformation group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L \subseteq S_4$$\end{document}L⊆S4 which is isomorphic to the dihedral group. Here, we extend the above findings to circular codes over a finite alphabet of even cardinality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\Sigma |=2n$$\end{document}|Σ|=2n for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \in {\mathbb {N}}$$\end{document}n∈N. We describe the corresponding group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_n$$\end{document}Ln using matrices and we investigate what classes of circular codes are split into equally sized equivalence classes under the natural equivalence relation induced by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_n$$\end{document}Ln. Surprisingly, this is not always the case. All results and constructions are illustrated by examples.