Tidjani NEGADI

 In this study, we once again use a set of Fibonacci-like sequences to examine the symmetries within the genetic code. This time, our focus is on the physiological state of the amino acids, considering them as charged, in contrast to our previous work where they were seen as neutral. In a pH environment around 7.4, there are four charged amino acids. We utilize the properties of our sequences to accurately describe the symmetries in the genetic code table. These include Rumer’s symmetry, the third-base symmetry and the “ideal” symmetry, along with the “supersymmetry”classification schemes. We also explore the special chemical structure of the amino acid proline, presenting two perspectives—shCherbak’s view and the Downes–Richardson view—which are included in the description of the above-mentioned symmetries. Our investigation also employs elementary modular arithmetic to precisely describe the chemical structure of proline, connecting the two views seamlessly. Finally, our Fibonacci-like sequences prove instrumental in quickly establishing the multiplet structure of non-standard versions of the genetic code. We illustrate this with an example, showcasing the efficiency of our method in unraveling the complex relationships within the genetic code.

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

The standard genetic code multiplet structure as well as the correct degeneracies, class by class, are all extracted from the (unique) number 23!, the order of the permutation group of 23 objects.

Journal Reference: Symmetry: Culture and Science, Vol. 18, 2-3, 149-160, 2007.  

This work aims at showing the relevance and the applications possibilities of the Fibonacci sequence, and also its q-deformed or "quantum" extension, in the study of the genetic code(s). First, after the presentation of a new formula, an indexed double Fibonacci sequence, comprising the first six Fibonacci numbers, is shown to describe the 20 amino acids multilets and their degeneracy as well as well as a characteristic pattern for the 61 meaningful codons. Next, the twenty amino acids, classified according to their increasing atom-number (carbon, nitrogen, oxygen and sulfur), exhibit several Fibonacci sequence patterns. Several mathematical relations are given, describing several atom-number patterns. Finally, a q-Fibonacci simple phenomenological model, with q a real deformation parameter, is used to describe, in a unified way, not only the standard genetic code, when q=1, but also all slight variations of the latter when q is near 1, as well as the case of the 21st amino acid (selenocysteine) and the 22nd one (pyrrolysine), also when q is near 1. As a by-product of this elementary model, we also show that, in the limit q=0, the number of amino acids reaches the value 6, in good agreement with old and still persistent claims that life, in its early development, could have used only a small number of amino acis. 

This is the presentation for "Symmetry Festival 2013", Delft, the Netherlands 2-7 Aug. 2013. Completing the corresponding publication.

The number of atoms in the four ribonucleotides uridine monophosphate (UMP), cytidine monophosphate (CMP), adenine monophosphate (AMP) and guanine monophosphate (GMP) is taken as a key parameter. A mathematical relation describing the condensation of the three basic sub-units a nucleobase, a ribose and a phosphate group, to form a ribonucleotide, is first obtained from this parameter. Next, the use of the latter and Euler's totient function is shown to lead to the atom number content of the 64 codons and also to Rakocevic's pattern "2x3456". Finally, selected sums of Fibonacci numbers are shown to lead to the nucleon number content of the amino acids in various degeneracy patterns , and also to the multiplet structure of the 20 amino acids as well as to the degeneracy.

In this paper, which is an extension of a very recent one, we show, in a first part, that starting from RNA, pore exactly, from the detailed atomic composition of its basic components, the four ribonucleotides UMP, CMP, AMP and GMP, full of Fibonacci numbers, we can derive the "One-Number Model" of the genetic code which was introduced by us some years ago, in 2007. In a second and original part, we consider the opposite way, that is, starting from the "One-Number Model", we can recover, in the detail, the atomic composition of the four ribonucleotides, from which we started, and the "circle is complete".

In this short paper, we present an interesting and welcome connection between two different approaches to the mathematical structure of the standard genetic code, we have considered in the last years. The first one relies on the use of the unique number 23! and the second is based on the atomic composition of the four ribonucleotides UMP, CMP, AMP and GMP, the building-blocks of RNA, where several Fibonacci numbers are seen to occur.