Comma-free Code Research Articles

Property based analysis: Optimality of RNY comma-free code versus circular code (X) after frameshift errors

The Standard Genetic Code (SGC) is a mapping between 64 codons and 20 natural amino acids plus a stop signal. The SGC is commonly considered to have been developed to lessen the consequences of translational blunders, such as the charging of erroneous amino acids during protein synthesis or a stop codon encounter. The Circular code X consists of 20 codons prevalent in the protein-coding genes of most species, including viruses, archaea, plasmids, bacteria, and eukaryotes. The Circular code X has remarkable error-correcting property, and its motifs are significantly abundant in genes, allowing for the detection and preservation of the proper reading frame. In this work, the hypothesis that the RNY Comma-free code is better optimized than the Circular code X at reducing the consequences of frameshift problems is put forward. The RNY code (a Self-complementary Comma-free code) consists of 16 codons observed statistically in gene sequences, where R stands for Purine and Y for Pyrimidine, and it has a nice error-correcting feature concerning the frameshift issue. We have employed the two previously developed score measures: code score and di-codon score, and then estimated the optimality of different codes based on the numerous physicochemical properties across all plausible frameshift mistakes. Later, we show that the RNY Comma-free code is better optimized than the Circular code X at reducing the effects of frameshift faults. Furthermore, some previously established results led us to infer that Circular code X is better optimized than the SGC in terms of reducing the impacts of frameshift errors. Accordingly, in terms of optimality following frameshift faults, the RNY Comma-free code outperforms the SGC. Our results support the Crick’s assertion that genetic codes evolved from archaic Comma-free codes, which may be more error-tolerant and hence more robust.

Generalized Systematic Comma-Free Code

In channels that introduce substitution, insertion, and deletion errors, one challenging problem for a code designer to overcome is to avoid false code-synchronization. In other words, the probability of a false codeword occurring should be minimized with appropriate code designs. In this paper, we propose a new class of systematic comma-free code called generalized $F(n,s,t)$ code for channels that are impaired by substitution, insertion, and deletion errors. We first prove that this code is a synchronous code and then we derive the probabilities of false synchronization when a substitution, insertion, or deletion error occurs. We compare the theoretical and simulation results under different channel parameters. We also analyze the factors affecting the false synchronization for each case. The proposed code gives a higher code efficiency and better choice in terms of false synchronization compared with the classical F codes.

Strong Comma-Free Codes in Genetic Information.

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.

Two-dimensional comma-free and cylindric codes

A two-dimensional code of pictures is defined as a set X⊆Σ⁎⁎ such that any picture over Σ is tilable in at most one way with pictures in X. It has been proved that it is undecidable whether a finite set of pictures is a code. Here we introduce two classes of picture codes: the comma-free codes and the cylindric codes, with the aim of generalizing the definitions of comma-free (or self-synchronizing) code and circular code of strings. The properties of these classes are studied and compared, in particular in relation to maximality and completeness. As a byproduct, we introduce self-covering pictures and study their periodicity issues.

A relation between trinucleotide comma-free codes and trinucleotide circular codes

The comma-free codes and circular codes are two important classes of codes in code theory and in genetics. Fifty years ago before the discovery of the genetic code, a trinucleotide (triletter) comma-free code was proposed for associating the codons of genes with the amino acids of proteins. More recently, in the last ten years, trinucleotide circular codes have been identified statistically in different genomes. Here, we identify a relation between these two classes of trinucleotide codes by constructing a hierarchy of comma-free and circular codes.

Properties of pure codes

In this paper we give some properties of pure codes and several characterizations on two-element pure codes. It is shown that a language is a solid code if and only if it is a comma-free code which is also a d-code. We also show that d-codes, intercodes and solid codes are pure codes.

Annihilators of bifix codes

A bifix code is a prefix code which is also a suffix code. The family of all bifix codes is the most important family of all codes. This family has been split into subfamilies such as the family of comma codes, the family of comma-free codes and the families of strict intercodes of index m, m ≥ 2. This paper is a study of the properties of the annihilators of a given bifix code. The types of annihilators we will consider are comma code, comma-free code and strict intercode of index m annihilators. We present the existence of each kind of annihilators we considered. We obtained several interesting properties together with two-word comma codes and strict intercodes of index 2.

Finite Completion of comma-free codes Part 1

This paper is the first step in the solution of the problem of finite completion of comma-free codes. We show that every finite comma-free code is included in a finite comma-free code of particular kind, which we called, for lack of a better term, canonical comma-free code. Certainly, finite maximal comma-free codes are always canonical. The final step of the solution which consists in proving further that every canonical comma-free code is completed to a finite maximal comma-free code, is intended to be published in a forthcoming paper.

Finite Completion of comma-free codes Part 2

This paper is a sequel to an earlier paper of the present author, in which it was proved that every finite comma-free code is embedded into a so-called (finite) canonical comma-free code. In this paper, it is proved that every (finite) canonical comma-free code is embedded into a finite maximal comma-free code, which thus achieves the conclusion that every finite comma-free code has finite completions.

Completing comma-free codes

We prove that for every regular comma-free code there exists a maximal comma-free code containing it which is still regular and, moreover, we can effectively locate such a completion. In particular, we can decide whether a given regular comma-free code is maximal.

SOME PROPERTIES OF MAXIMAL COMMA-FREE CODES

A code $A$ over a finite alphabet $X$ is a comma-free code if $A^2\cap X^+AX^+=\emptyset$, where $X$ is a finite alphabet containing more than one letter. This paper is a study of some algebraic properties of finite maximal comma-free codes. We give several characterizations on two-element comma-free codes and maximal comma-free codes. Let $X_n^m= X^n\cup X^{n+1}\cup \cdots \cup X^m$. We prove that for $n \ge 3$ , a maximal comma-free code in $X^n$ is a maximal comma-free code in the region $X_1^m\cup X^n$, $m < n/2$. We also obtain that for $X = \{a,b\}$, a maximal comma-free code in $X^3$ is a maximal comma-free code; a maximal comma-free code in $X^4$ is a maximal comma-free code in $X_1^4$; for every $n\ge 4$, there is a maximal comma-free code in $X^n$ which is not a maximal comma-free code.

The syntactic monoid of the semigroup generated by a comma-free code

A sequence of lemmas leads to a two-fold characterisation of the syntactic monoid in the title. Some alternatives as well as special cases, in particular when the code consists of a singleton, are considered.

Comma-free and synchronizable codes

Shepherd (1981. Proc. natn. Acad. Sci. U.S.A. 78, 1596-1600), Clarke (1982. J. theor. Biol. 99, 397-403), Crick et al. (1976. Origins of Life 7, 389) have put forward the possibility that there was a code, ancestral to the existing genetic code, which was non-degenerate and comma-free. The point of a comma-free code, as first proposed in Crick et al. (1957), is that it provides immunity from frameshift reading errors. In this paper it is argued that if frameshift error immunity is to be used as a selection factor in guiding the search for a hypothetical ancestral code, then the appropriate mathematical formulation of this characteristic is not comma-freeness but synchronizability. The calling card of the comma-free notion is that it results in a limit of 20 on the number of sense codons when applied to the genetic coding system. But so does synchronizability, which includes comma-freeness as a special case. Further, using synchronizability as a constraint instead of comma-freeness yields roughly 3000 times as many potential models for an ancestral code. In particular, the RNY model becomes more compatible with immunity from frameshift error as an evolutionary constraint.

Finite State Models in the Study of Comma-Free Codes

The definition of comma-free codes is extended to be more comprehensive and include codes with variable length codewords and codes with an unlimited number of codewords. Finite state recognizers are used to represent commafree codes in a compact form, making available the well established methods of manipulating and analysing finite state models. An efficient algorithm is presented to test the comma-freeness of any given code represented by a finite state recognizer. It is well known, that any maximal comma-free code with codewords of odd fixed length can be achieved. Although upper limits have been given for even fixed length it has not been established which comma-free codes of this nature can achieve the upper limit. Compact representations of comma-free binary codes are given for variable length codewords with maximum even fixed lengths (6, 8 and 10) and in one case (length not greater than 6) it is seen that the corresponding upper limit for fixed lengths can be exceeded.

Hypothetical stages in the mutational evolution of the present-day genetic code from its RNY, comma-free ancestor

Abstract The primitive comma-free genetic code may have had 16 triplets of the form RNY, where R = purine, N = purine or pyrimidine, and Y = pyrimidine, specifying eight (present-day) amino acids. Calculations reveal that in this primitive code all transition changes (A↔G, C↔U) are either silent or missense i.e. result in the same or another one of these particular eight amino acids. There are no single transitions to non-RNY codons. Single transversions in the primitive codons can, individually, generate new (present-day) codons for four or eight amino acids. Present-day glutamine, tryptophan and stop (UGA, UAA, UAG) codons cannot be so derived., by single transversions, from any of the eight primitive codons. The modern initiation codons, AUG and GUG, can however be generated by both C → G and U → G single transversions in primitive codons. Overall, a total of 32 modern sense codons, not represented in the primitive RNY code, can be derived from this code by single transversions. Many modern codons, including all those not generated by single transversions in the primitive code, can also be produced by either of the two types of frameshift possible in runs of U- or C-rich primitive codons. Present-day stop codons are generated by +1 (-2) type frameshifts in U-rich primitive runs; AUG and GUG initiation codons are produced by the other type, +2 (-1), frameshifts in U-rich runs.

On maximal comma-free codes (Corresp.)

In this correspondence a binary maximal comma-free code for words of length 10 and the search technique used to construct it are described.

Comma-Free Ternary Triorthogonal Code

This paper shows the existence of a ternary triorthogonal code and a comma-free ternary triorthogonal code. Some theorems are proved for their existence and some algorithms for generating these codes are developed. For a dictionary of size $D = 3^{n + 1} $ and word length $L = 3^n $, the former code exists for any positive integer n and the latter code exists for $n > 1$. The ternary elements $\{ 0,1,2\} $ used in the codes are isomorphic, under the one-to-one correspondence, to the triphase vectors $\{ e^{(2\pi i/3 )r} ,r = 0,1,2\}$. The modulo 3 addition of $\{ 0,1,2\} $ and the triphase vector multiplication are also isomorphic under the one-to-one correspondence.