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Comma-free codes constitute a class of circular codes, which has been widely studied, in particular by Golomb et al. (Biologiske Meddelelser, Kongelige Danske Videnskabernes Selskab 23:1-34, 1958a, Can J Math 10:202-209, 1958b), Michel et al. (Comput Math Appl 55:989-996, 2008a, Theor Comput Sci 401:17-26, 2008b, Inf Comput 212:55-63, 2012), Michel and Pirillo (Int J Comb 2011:659567, 2011), and Fimmel and Strüngmann (J Theor Biol 389:206-213, 2016). Based on a recent approach using graph theory to study circular codes Fimmel et al. (Philos Trans R Soc 374:20150058, 2016), a new class of circular codes, called strong comma-free codes, is identified. These codes detect a frameshift during the translation process immediately after a reading window of at most two nucleotides. We describe several combinatorial properties of strong comma-free codes: enumeration, maximality, self-complementarity and [Formula: see text]-property (comma-free property in all the three possible frames). These combinatorial results also highlight some new properties of the genetic code and its evolution. Each amino acid in the standard genetic code is coded by at least one strong comma-free code of size 1. There are 9 amino acids [Formula: see text] among 20 such that for each amino acid from S, its synonymous trinucleotide set (excluding the necessary periodic trinucleotides [Formula: see text]) is a strong comma-free code. The primeval comma-free RNY code of Eigen and Schuster (Naturwissenschaften 65:341-369, 1978) is a self-complementary [Formula: see text]-code of size 16. Furthermore, it is the union of two strong comma-free codes of size 8 which are complementary to each other.

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., (Copyright © 2014 Elsevier Ltd. All rights reserved.)

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. [ABSTRACT FROM AUTHOR]