An Extended RNA Code and its Relationship to the Standard Genetic Code: An Algebraic and Geometrical Approach

Significance of genetic code module structure in gene expression and GC content enhancement in RNA sequences

The existent algebraic models of the genetic code contribute to the understanding of the physio-chemical characteristics of the amino acids. However, the process of translating a gene into a phenotype is highly complex. Moreover, the intricacy of gene expression gets further multiplied due to the biases in the codon usage. This paper explores an algebraic structure called module on the set of codons as well as on that of RNA sequences. We study the potential implications of these structures on gene expression and the GC content of an RNA sequence. The base order {C,U,G,A} appears to possess greater biological significance than many of the orders previously studied. We have developed a novel algorithm to generate RNA sequences with high GC content, aiming to enhance the thermostability of biomolecules. The insights gained from this investigation may have applications in biomolecular modeling and docking, protein engineering, drug development, and related fields.

Groups of Symmetries of the Two Classes of Synthetases in the Four-Dimensional Hypercubes of the Extended Code Type II

Aminoacyl-tRNA synthetases (aaRSs) originated from an ancestral bidirectional gene (mirror symmetry), and through the evolution of the genetic code, the twenty aaRSs exhibit a symmetrical distribution in a 6-dimensional hypercube of the Standard Genetic Code. In this work, we assume a primeval RNY code and the Extended Genetic RNA code type II, which includes codons of the types YNY, YNR, and RNR. Each of the four subsets of codons can be represented in a 4-dimensional hypercube. Altogether, these 4 subcodes constitute the 6-dimensional representation of the SGC. We identify the aaRSs symmetry groups in each of these hypercubes. We show that each of the four hypercubes contains the following sets of symmetries for the two known Classes of synthetases: RNY: dihedral group of order 4; YNY: binary group; YNR: amplified octahedral group; and RNR: binary group. We demonstrate that for each hypercube, the group of symmetries in Class 1 is the same as the group of symmetries in Class 2. The biological implications of these findings are discussed.

Symmetrical distributions of aminoacyl-tRNA synthetases during the evolution of the genetic code

In this work, we formulate the following question: How the distribution of aminoacyl-tRNA synthetases (aaRSs) went from an ancestral bidirectional gene (mirror symmetry) to the symmetrical distribution of aaRSs in a six-dimensional hypercube of the Standard Genetic Code (SGC)? We assume a primeval RNY code, two Extended Genetic RNA codes type 1 and 2, and the SGC. We outline the types of symmetries of the distribution of aaRSs in each code. The symmetry groups of aaRSs in each code are described, until the symmetries of the SGC display a mirror symmetry. Considering both Extended RNA codes the 20 aaRSs were already present before the Last Universal Ancestor. These findings reveal intricacies in the diversification of aaRSs accompanied by the evolution of the genetic code.

On the Importance of Asymmetry in the Phenotypic Expression of the Genetic Code upon the Molecular Evolution of Proteins

The standard genetic code (SGC) is a mapping between the 64 possible arrangements of the four RNA nucleotides (C, A, U, G) into triplets or codons, where 61 codons are assigned to a specific amino acid and the other three are stop codons for terminating protein synthesis. Aminoacyl-tRNA synthetases (aaRSs) are responsible for implementing the SGC by specifically amino-acylating only its cognate transfer RNA (tRNA), thereby linking an amino acid with its corresponding anticodon triplets. tRNAs molecules bind each codon with its anticodon. To understand the meaning of symmetrical/asymmetrical properties of the SGC, we designed synthetic genetic codes with known symmetries and with the same degeneracy of the SGC. We determined their impact on the substitution rates for each amino acid under a neutral model of protein evolution. We prove that the phenotypic graphs of the SGC for codons and anticodons for all the possible arrangements of nucleotides are asymmetric and the amino acids do not form orbits. In the symmetrical synthetic codes, the amino acids are grouped according to their codonicity, this is the number of triplets encoding a given amino acid. Both the SGC and symmetrical synthetic codes exhibit a probability of occurrence of the amino acids proportional to their degeneracy. Unlike the SGC, the synthetic codes display a constant probability of occurrence of the amino acid according to their codonicity. The asymmetry of the phenotypic graphs of codons and anticodons of the SGC, has important implications on the evolutionary processes of proteins.

A neutral evolution test derived from a theoretical amino acid substitution model

A neutral evolution model that explicitly considers codons, amino acids, and the degeneracy of the genetic code is developed. The model is built from nucleotides up to amino acids, and it represents a refinement of the neutral theory of molecular evolution. The model is based on a stochastic process that leads to a stationary probability distribution of amino acids. The latter is used as a neutral test of evolution. We provide some examples for assessing the neutrality test for a small set of protein sequences. The Jukes-Cantor model is generalized to deal with amino acids and it is compared with our neutral model, along with the empirical BLOSUM62 substitution model. The neutral test provides a baseline to which the evolution of any protein can be analyzed, and it clearly helps in discerning putative amino acids with unexpected frequencies that might be under positive or negative selection. Our model and neutral test are as universal as the standard genetic code.

Symmetrical Properties of Graph Representations of Genetic Codes: From Genotype to Phenotype

It has long been claimed that the mitochondrial genetic code possesses more symmetries than the Standard Genetic Code (SGC). To test this claim, the symmetrical structure of the SGC is compared with noncanonical genetic codes. We analyzed the symmetries of the graphs of codons and their respective phenotypic graph representation spanned by the RNY (R purines, Y pyrimidines, and N any of them) code, two RNA Extended codes, the SGC, as well as three different mitochondrial genetic codes from yeast, invertebrates, and vertebrates. The symmetry groups of the SGC and their corresponding phenotypic graphs of amino acids expose the evolvability of the SGC. Indeed, the analyzed mitochondrial genetic codes are more symmetrical than the SGC.

A Proposal of the Ur-proteome

Herein we outline a plausible proteome, encoded by assuming a primeval RNY genetic code. We unveil the primeval phenotype by using only the RNA genotype; it means that we recovered the most ancestral proteome, mostly made of the 8 amino acids encoded by RNY triplets. By looking at those fragments, it is noticeable that they are positioned, not at catalytic sites, but in the cofactor binding sites. It implies that the stabilization of a molecule appeared long before its catalytic activity, and therefore the Ur-proteome comprised a set of proteins modules that corresponded to Cofactor Stabilizing Binding Sites (CSBSs), which we call the primitive bindome. With our method, we reconstructed the structures of the "first protein modules" that Sobolevsky and Trifonov (2006) found by using only RMSD. We also examine the probable cofactors that bound to them. We discuss the notion of CSBSs as the first proteins modules in progenotes in the context of several proposals about the primitive forms of life.

Phenotypic Graphs and Evolution Unfold the Standard Genetic Code as the Optimal

In this work, we explicitly consider the evolution of the Standard Genetic Code (SGC) by assuming two evolutionary stages, to wit, the primeval RNY code and two intermediate codes in between. We used network theory and graph theory to measure the connectivity of each phenotypic graph. The connectivity values are compared to the values of the codes under different randomization scenarios. An error-correcting optimal code is one in which the algebraic connectivity is minimized. We show that the SGC is optimal in regard to its robustness and error-tolerance when compared to all random codes under different assumptions.

A unified model of the standard genetic code

The Rodin-Ohno (RO) and the Delarue models divide the table of the genetic code into two classes of aminoacyl-tRNA synthetases (aaRSs I and II) with recognition from the minor or major groove sides of the tRNA acceptor stem, respectively. These models are asymmetric but they are biologically meaningful. On the other hand, the standard genetic code (SGC) can be derived from the primeval RNY code (R stands for purines, Y for pyrimidines and N any of them). In this work, the RO-model is derived by means of group actions, namely, symmetries represented by automorphisms, assuming that the SGC originated from a primeval RNY code. It turns out that the RO-model is symmetric in a six-dimensional (6D) hypercube. Conversely, using the same automorphisms, we show that the RO-model can lead to the SGC. In addition, the asymmetric Delarue model becomes symmetric by means of quotient group operations. We formulate isometric functions that convert the class aaRS I into the class aaRS II and vice versa. We show that the four polar requirement categories display a symmetrical arrangement in our 6D hypercube. Altogether these results cannot be attained, neither in two nor in three dimensions. We discuss the present unified 6D algebraic model, which is compatible with both the SGC (based upon the primeval RNY code) and the RO-model.

On the Uniqueness of the Standard Genetic Code

In this work, we determine the biological and mathematical properties that are sufficient and necessary to uniquely determine both the primeval RNY (purine-any base-pyrimidine) code and the standard genetic code (SGC). These properties are: the evolution of the SGC from the RNY code; the degeneracy of both codes, and the non-degeneracy of the assignments of aminoacyl-tRNA synthetases (aaRSs) to amino acids; the wobbling property; the consideration that glycine was the first amino acid; the topological and symmetrical properties of both codes.

Codon Distribution in Error-Detecting Circular Codes

In 1957, Francis Crick et al. suggested an ingenious explanation for the process of frame maintenance. The idea was based on the notion of comma-free codes. Although Crick’s hypothesis proved to be wrong, in 1996, Arquès and Michel discovered the existence of a weaker version of such codes in eukaryote and prokaryote genomes, namely the so-called circular codes. Since then, circular code theory has invariably evoked great interest and made significant progress. In this article, the codon distributions in maximal comma-free, maximal self-complementary C3 and maximal self-complementary circular codes are discussed, i.e., we investigate in how many of such codes a given codon participates. As the main (and surprising) result, it is shown that the codons can be separated into very few classes (three, or five, or six) with respect to their frequency. Moreover, the distribution classes can be hierarchically ordered as refinements from maximal comma-free codes via maximal self-complementary C3 codes to maximal self-complementary circular codes.

Maximal dinucleotide comma-free codes

The problem of retrieval and maintenance of the correct reading frame plays a significant role in RNA transcription. Circular codes, and especially comma-free codes, can help to understand the underlying mechanisms of error-detection in this process. In recent years much attention has been paid to the investigation of trinucleotide circular codes (see, for instance, Fimmel et al., 2014; Fimmel and Strüngmann, 2015a; Michel and Pirillo, 2012; Michel et al., 2012, 2008), while dinucleotide codes had been touched on only marginally, even though dinucleotides are associated to important biological functions. Recently, all maximal dinucleotide circular codes were classified (Fimmel et al., 2015; Michel and Pirillo, 2013). The present paper studies maximal dinucleotide comma-free codes and their close connection to maximal dinucleotide circular codes. We give a construction principle for such codes and provide a graphical representation that allows them to be visualized geometrically. Moreover, we compare the results for dinucleotide codes with the corresponding situation for trinucleotide maximal self-complementary C(3)-codes. Finally, the results obtained are discussed with respect to Crick׳s hypothesis about frame-shift-detecting codes without commas.

Symmetrical and Thermodynamic Properties of Phenotypic Graphs of Amino Acids Encoded by the Primeval RNY Code

The 12 different types of graphs of the 8 amino acids encoded by the presumably primeval RNY code are derived. The symmetry groups of these graphs are analyzed and coincide with the corresponding values of polar requirement for each amino acid. The symmetry groups at the codon level are partially carried over as a group or subgroup at the amino acid level. Measures of centrality of the 12 graphs indicate that all amino acids were equally relevant irrespective of its chronological order of its appearance. The elimination of any amino acid would be strongly selected against and therefore the genetic code at this stage was already frozen.

Three-Dimensional Algebraic Models of the tRNA Code and 12 Graphs for Representing the Amino Acids

Three-dimensional algebraic models, also called Genetic Hotels, are developed to represent the Standard Genetic Code, the Standard tRNA Code (S-tRNA-C), and the Human tRNA code (H-tRNA-C). New algebraic concepts are introduced to be able to describe these models, to wit, the generalization of the 2n-Klein Group and the concept of a subgroup coset with a tail. We found that the H-tRNA-C displayed broken symmetries in regard to the S-tRNA-C, which is highly symmetric. We also show that there are only 12 ways to represent each of the corresponding phenotypic graphs of amino acids. The averages of statistical centrality measures of the 12 graphs for each of the three codes are carried out and they are statistically compared. The phenotypic graphs of the S-tRNA-C display a common triangular prism of amino acids in 10 out of the 12 graphs, whilst the corresponding graphs for the H-tRNA-C display only two triangular prisms. The graphs exhibit disjoint clusters of amino acids when their polar requirement values are used. We contend that the S-tRNA-C is in a frozen-like state, whereas the H-tRNA-C may be in an evolving state.

Genetic hotels for the standard genetic code: evolutionary analysis based upon novel three-dimensional algebraic models

Herein, we rigorously develop novel 3-dimensional algebraic models called Genetic Hotels of the Standard Genetic Code (SGC). We start by considering the primeval RNA genetic code which consists of the 16 codons of type RNY (purine-any base-pyrimidine). Using simple algebraic operations, we show how the RNA code could have evolved toward the current SGC via two different intermediate evolutionary stages called Extended RNA code type I and II. By rotations or translations of the subset RNY, we arrive at the SGC via the former (type I) or via the latter (type II), respectively. Biologically, the Extended RNA code type I, consists of all codons of the type RNY plus codons obtained by considering the RNA code but in the second (NYR type) and third (YRN type) reading frames. The Extended RNA code type II, comprises all codons of the type RNY plus codons that arise from transversions of the RNA code in the first (YNY type) and third (RNR) nucleotide bases. Since the dimensions of remarkable subsets of the Genetic Hotels are not necessarily integer numbers, we also introduce the concept of algebraic fractal dimension. A general decoding function which maps each codon to its corresponding amino acid or the stop signals is also derived. The Phenotypic Hotel of amino acids is also illustrated. The proposed evolutionary paths are discussed in terms of the existing theories of the evolution of the SGC. The adoption of 3-dimensional models of the Genetic and Phenotypic Hotels will facilitate the understanding of the biological properties of the SGC.

On the evolution of the standard genetic code: vestiges of critical scale invariance from the RNA world in current prokaryote genomes

Herein two genetic codes from which the primeval RNA code could have originated the standard genetic code (SGC) are derived. One of them, called extended RNA code type I, consists of all codons of the type RNY (purine-any base-pyrimidine) plus codons obtained by considering the RNA code but in the second (NYR type) and third (YRN type) reading frames. The extended RNA code type II, comprises all codons of the type RNY plus codons that arise from transversions of the RNA code in the first (YNY type) and third (RNR) nucleotide bases. In order to test if putative nucleotide sequences in the RNA World and in both extended RNA codes, share the same scaling and statistical properties to those encountered in current prokaryotes, we used the genomes of four Eubacteria and three Archaeas. For each prokaryote, we obtained their respective genomes obeying the RNA code or the extended RNA codes types I and II. In each case, we estimated the scaling properties of triplet sequences via a renormalization group approach, and we calculated the frequency distributions of distances for each codon. Remarkably, the scaling properties of the distance series of some codons from the RNA code and most codons from both extended RNA codes turned out to be identical or very close to the scaling properties of codons of the SGC. To test for the robustness of these results, we show, via computer simulation experiments, that random mutations of current genomes, at the rates of 10(-10) per site per year during three billions of years, were not enough for destroying the observed patterns. Therefore, we conclude that most current prokaryotes may still contain relics of the primeval RNA World and that both extended RNA codes may well represent two plausible evolutionary paths between the RNA code and the current SGC.