Search Results (10)

By imposing finite order constraints on Fibonacci anyon braid relations, we construct the finite quotient $G={\mathbb{Z}}_5⋊2I$, where $2I$ is the binary icosahedral group. The Gröbner basis decomposition of its $SL(2,\mathbb{C})$ character variety yields elliptic curves whose L-function derivatives $L^{\prime }(E,1)$ remarkably match fundamental biological structural ratios. Specifically, we demonstrate that the Birch–Swinnerton-Dyer conjecture’s central quantity: the derivative $L^{\prime }(E,1)$ of the L-function at 1 encodes critical cellular geometries: the crystalline B-DNA pitch-to-diameter ratio ($L^{\prime }(E,1)=1.730$ matching $34\AA /20\AA =1.70$), the B-DNA pitch to major groove width ($L^{\prime }=1.58$) and, additionally, the fundamental cytoskeletal scaling relationship where $L^{\prime }(E,1)=3.570\approx 25/7$, precisely matching the microtubule-to-actin diameter ratio. This pattern extends across the hierarchy ${\mathbb{Z}}_5⋊2P$ with $2P\in \{2O,2T,2I\}$ (binary octahedral, tetrahedral, icosahedral groups), where character tables of $2O$ explain genetic code degeneracies while $2T$ yields microtubule ratios. The convergence of multiple independent mathematical pathways on identical biological values suggests that evolutionary optimization operates under deep arithmetic-geometric constraints encoded in elliptic curve L-functions. Our results position the BSD conjecture not merely as abstract number theory, but as encoding fundamental organizational principles governing cellular architecture. The correspondence reveals arithmetic geometry as the mathematical blueprint underlying major biological structural systems, with Gross–Zagier theory providing the theoretical framework connecting quantum topology to the helical geometries that are essential for life. Full article

Synonymous mutations result from the degeneracy of the genetic code. Most amino acids are encoded by two or more codons, and mutations that change a codon to another synonymous codon do not change the amino acid in the gene product. Historically, such mutations have been considered silent because they were assumed to have no to very little impact. However, research in the last few decades has produced several examples where synonymous mutations play important roles. These include optimizing expression by enhancing translation initiation and accelerating or decelerating translation elongation via codon usage and mRNA secondary structures, stabilizing mRNA molecules and preventing their breakdown before translation, and faulty protein folding or increased degradation due to enhanced ubiquitination and suboptimal secretion of proteins into the appropriate cell compartments. Some consequences of synonymous mutations, such as mRNA stability, can lead to different outcomes in prokaryotes and eukaryotes. Despite these examples, the significance of synonymous mutations in evolution and in causing disease in comparison to nonsynonymous mutations that do change amino acid residues in proteins remains controversial. Whether the molecular mechanisms described by which synonymous mutations affect organisms can be generalized remains poorly understood and warrants future research in this area. Full article

In this work, we present a new way of studying the mathematical structure of the genetic code. This study relies on the use of mathematical computations involving five Fibonacci-like sequences; a few of their “seeds” or “initial conditions” are chosen according to the chemical and physical data of the three amino acids serine, arginine and leucine, playing a prominent role in a recent symmetry classification scheme of the genetic code. It appears that these mathematical sequences, of the same kind as the famous Fibonacci series, apart from their usual recurrence relations, are highly intertwined by many useful linear relationships. Using these sequences and also various sums or linear combinations of them, we derive several physical and chemical quantities of interest, such as the number of total coding codons, 61, obeying various degeneracy patterns, the detailed number of H/CNOS atoms and the integer molecular mass (or nucleon number), in the side chains of the coded amino acids and also in various degeneracy patterns, in agreement with those described in the literature. We also discover, as a by-product, an accurate description of the very chemical structure of the four ribonucleotides uridine monophosphate (UMP), cytidine monophosphate (CMP), adenosine monophosphate (AMP) and guanosine monophosphate (GMP), the building blocks of RNA whose groupings, in three units, constitute the triplet codons. In summary, we find a full mathematical and chemical connection with the “ideal sextet’s classification scheme”, which we alluded to above, as well as with others—notably, the Findley–Findley–McGlynn and Rumer’s symmetrical classifications. Full article

The genetic code evolved around the reading of the tRNA anticodon on the primitive ribosome, and tRNA-34 wobble and tRNA-37 modifications coevolved with the code. We posit that EF-Tu, the closing mechanism of the 30S ribosomal subunit, methylation of wobble U34 at the 5-carbon and suppression of wobbling at the tRNA-36 position were partly redundant and overlapping functions that coevolved to establish the code. The genetic code devolved in evolution of mitochondria to reduce the size of the tRNAome (all of the tRNAs of an organism or organelle). “Superwobbling” or four-way wobbling describes a major mechanism for shrinking the mitochondrial tRNAome. In superwobbling, unmodified wobble tRNA-U34 can recognize all four codon wobble bases (A, G, C and U), allowing a single unmodified tRNA-U34 to read a 4-codon box. During code evolution, to suppress superwobbling in 2-codon sectors, U34 modification by methylation at the 5-carbon position appears essential. As expected, at the base of code evolution, tRNA-37 modifications mostly related to the identity of the adjacent tRNA-36 base. TRNA-37 modifications help maintain the translation frame during elongation. Full article

Nirenberg’s genetic code chart shows a profound correspondence between codons and amino acids. The aim of this article is to try to explain the primordial formation of the codon degeneracy. It remains a puzzle how informative molecules arose from the supposed prebiotic random sequences. If introducing an initial driving force based on the relative stabilities of triplex base pairs, the prebiotic sequence evolution became innately nonrandom. Thus, the primordial assignment of the 64 codons to the 20 amino acids has been explained in detail according to base substitutions during the coevolution of tRNAs with aaRSs; meanwhile, the classification of aaRSs has also been explained. Full article

We find that the degeneracies and many peculiarities of the DNA genetic code may be described thanks to two closely related (fivefold symmetric) finite groups. The first group has signature $G={\mathbb{Z}}_5⋊H$ where $H={\mathbb{Z}}_2.S_4\cong 2O$ is isomorphic to the binary octahedral group $2O$ and $S_4$ is the symmetric group on four letters/bases. The second group has signature $G={\mathbb{Z}}_5⋊GL(2,3)$ and points out a threefold symmetry of base pairings. For those groups, the representations for the 22 conjugacy classes of G are in one-to-one correspondence with the multiplets encoding the proteinogenic amino acids. Additionally, most of the 22 characters of G attached to those representations are informationally complete. The biological meaning of these coincidences is discussed. Full article

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. Full article

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. Full article

First, mathematical formulae faithfully describing the distributions of amino acids and codons and reproducing the degeneracies in the various known genetic codes, including the standard genetic code, are constructed, by hand. Second, we summarize another mathematical approach relying on the use of q-deformations to describe these same genetic codes, and add a new application not considered before. Third, by considering these same genetic codes, we find, qualitatively, that an inverse symmetry-information relationship exists. Full article

The similarity patterns of the genetic code result from similar codons encoding similar messages. We develop a new mathematical model to analyze these patterns. The physicochemical characteristics of amino acids objectively quantify their differences and similarities; the Hamming metric does the same for the 64 codons of the codon set. (Hamming distances equal the number of different codon positions: AAA and AAC are at 1-distance; codons are maximally at 3-distance.) The CodonPolytope, a 9-dimensional geometric object, is spanned by 64 vertices that represent the codons and the Euclidian distances between these vertices correspond one-to-one with intercodon Hamming distances. The CodonGraph represents the vertices and edges of the polytope; each edge equals a Hamming 1-distance. The mirror reflection symmetry group of the polytope is isomorphic to the largest permutation symmetry group of the codon set that preserves Hamming distances. These groups contain 82,944 symmetries. Many polytope symmetries coincide with the degeneracy and similarity patterns of the genetic code. These code symmetries are strongly related with the face structure of the polytope with smaller faces displaying stronger code symmetries. Splitting the polytope stepwise into smaller faces models an early evolution of the code that generates this hierarchy of code symmetries. The canonical code represents a class of 41,472 codes with equivalent symmetries; a single class among an astronomical number of symmetry classes comprising all possible codes. Full article