A Mathematical Model for the Genetic Code(s) Based on Fibonacci Numbers and their q-Analogues

Abstract

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 multiplets and their degeneracy 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 various 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 known slight variations of this latter, when q~1, as well as the case of the 21 st amino acid (selenocysteine) and the 22nd one (pyrrolysine), also when q~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 stating that life, in its early development, could have used only a small number of amino acids.

The Genetic Codes: Mathematical Formulae and an Inverse Symmetry-Information Relationship

“…Also, he gave no explicit mathematical formula for the supposed arithmetical progression for the frequencies, in both cases (even and odd multiplets). In [6], we have written the frequencies for the even multiplets, mentioned above, as the simple arithmetic progression 8 − 2k (k = 1, 2, 3), which gives 6, 4, and 2. We have therefore that the frequencies-3, 5, and 9-are in accordance with the geometrical progression 2 k + 1 when the even multiplets-6, 4, and 2-are inversely ordered by the arithmetical progression 8 − 2k, for k = 1, 2, 3.…”

“…Note also that, for the four most symmetric Figure 2. The action of the transformations R 3 , R 6 , and R 8 on the codon matrix C (see the text and [20]). The set M 1 is indicated in light grey, as in Figure 1.…”

“…The blue color corresponds to the substitutions {U↔G, A↔C}, the red color to the substitutions {A↔U, C↔G} and the green color to the substitutions to {U↔C, A↔G}; all the substitutions occur at the third base. For R 6 and R 8 , the codons shown in black have their position strictly unchanged. The codons corresponding to the stop codons are underlined.…”

“…In spite of this difficult problem, we have nonetheless tried, through the recent past years, to make some modest contributions to find mathematical models which, if not derived from "first principles", at least could faithfully reproduce the experimental data for the standard genetic code [3][4][5] and, to some extent, its important variant the Vertebrate Mitochondrial genetic code [4] (one could also find in [3][4][5] reference to other older works by ourselves). Recently, we have used the formalism of q-deformations (see Section 3) to study the case of the various today known genetic code variants [6,7]. We also constructed, by hand, exact mathematical expressions, relying on the use of geometrical and arithmetical progressions describing the distributions of amino acids and codons and reproducing faithfully the degeneracies for the standard genetic code [6], and the Vertebrate Mitochondrial Code [7].…”

“…Recently, we have used the formalism of q-deformations (see Section 3) to study the case of the various today known genetic code variants [6,7]. We also constructed, by hand, exact mathematical expressions, relying on the use of geometrical and arithmetical progressions describing the distributions of amino acids and codons and reproducing faithfully the degeneracies for the standard genetic code [6], and the Vertebrate Mitochondrial Code [7]. Here, in this new contribution, we go further and construct exact mathematical expressions for each one of all the genetic code variants which also reproduce, faithfully, the distribution of amino acids and codons as well as the degeneracies.…”