DOI: 10.14704/nq.2015.13.3.850

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

Tidjani Négadi


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 21st 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.


genetic code; Fibonacci numbers; degeneracy; atom-numbers; q-Fibonacci numbers; genetic codes

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Basin SL. The Fibonacci Sequence as it appears in Nature. The Fibonacci Quarterly 1963; 1(1): 53-56.

Beleza Yamagishi ME, Shimabukuro AI. Nucleotide Frequencies in Human Genome and Fibonacci Numbers. Bul.

Math. Biol. 2008; 70(3): 643-653.

Böck A, Forchhammer K, Heider J, Leinfelder W, Sawers G, Veprek B, Zinoni F. Selenocysteine: the 21st amin acid. Mol. Microbiol. 1991; 5: 515-520.

Carlitz L. Fibonacci Notes 3: q-Fibonacci Numbers. The Fibonacci Quarterly 1974; 12: 317-322.

Cohn JHE. On square Fibonacci numbers. Jour. Lond. Math. Soc. 1964; 39: 537-540.

Coldea R, Tennant DA, Wheeler EM, Wawrzynska E, Prabhakaran D, Telling M, Habicht K, Smeibidl P, Kiefer K.

Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry. Science 2010;

: 177-180. https://www.sni


Crick FHC. The Origin of the Genetic Code. J. Mol. Biol. 1968; 38: 367-379.

Di Giulio M. An Extension of the Coevolution Theory of the Genetic Code. Biology Direct 2008; 3: 37-57.

Erdmann VA, Barciszewski J. 2011: 50th Anniversary of the Discovery of the Genetic Code. Angew. Chem. Int. Ed. 2011; 50: 9546-9552.

Gavaudan P. The genetic code and the origin of life. In Chemical Evolution and the origin of life. North-Holland Publishing Company: Amsterdam, The Netherlands, 1971.

Hartman H, Smith TF. The Evolution of the Ribosome and the Genetic Code. Life 2014; 4: 227-249.

Higgs PG. A four-column theory for the origin of the genetic code: tracing the evolutionary pathways that gave rise to an optimized code. Biology Direct 2009; 4: 16.

Higgs PG, Pudritz RE. A Thermodynamic Basis for Prebiotic Amino acid Synthesis and the Nature of the First Genetic Code. Astrobiology 2009; 9(5): 483-490.

Jackson FH. A Basic Sine and Cosine with Symbolical Solutions of Certain Differential Equations. Proc. Edin. Math.Soc. 1904; 22: 28-39.

Kibler M, Négadi T. On the q-analogue of the Hydrogen Atom. J. Phys. A Math. Gen. 1991; 24: 5283. (arXiv:quant-ph/0408151v1)

Knott R. Fibonacci Numbers.


Linage G, Montoya F, Sarmiento A, Showalter K, Parmananda P. Fibonacci Order in the Period-Doubling Cascade

to Chaos. Phys. Lett. A 2006; 359: 638-639.

Macfarlane A. On q-analogues of the Quantum Harmonic Oscillator and the Quantum Group SU(2)q . J. Phys. A Math. Gen. 1989; 22: 4581.

Mäkelä T, Annila A. Natural patterns of energy dispersal. Physics of Life Reiew 2010; 7: 477-498.

Mitchison GJ. Phyllotaxis and the Fibonacci Series. Science 1977; 196: 270-275.

Négadi T. The Genetic Code Invariance: When Euler and Fibonacci Meet. Symmetry: Culture and Science 2014; 25(3): 261-278. (arXiv:1406.6092v1 [q-bio.OT].)

Négadi T. The Genetic Code Multiplet Structure, in One Number. Symmetry: Culture and Science 2007; 18(2-3): 149-160. (arXiv:0707.2011v1 [q-bio.OT].)

Négadi T. The Genetic Code Degeneracy and the Amino Acids Chemical Composition are Connected. NeuroQuantology 2009; 7(1): 181-187. (arXiv:0903.4131v1 [q-bio.OT].)

Négadi T. Work in progress (2015).

Négadi T, Kibler M. A q-deformed Aufbau Prinzip. J. Phys. A Math. Gen. 1992; 25: L157-L160. (arXiv:quant-ph/0409177v1)

Nirenberg M, Leder P, Bernfield M, Brimacombe R, Trupin J, Rottman F, O’Neal C. NA Codewords and Protein

Synthesis, VII. On the General Nature of the RNA Code. Proc. Natl. Acad. Sci. USA 1965; 53(5): 1161-1168.

Parker ET, Cleaves HJ, Dworkin JP, Glavin DP, Callahan M, Aubrey A, Lazcano A, Bada JL. Primordial synthesis of amines and amino acids in a 1958 Miller H2S-rich spark discharge experiment. Procs. Natl Acad Sci USA 2011; 108(14): 5526-5531.

Rakočević MM. The Genetic Code as a Golden Mean Determined System. Biosystems 1998; 46(3): 283-291.

Shechtman D, Blech I, Gratias D, Cahn, JW. Metallic Phase with Long Orientational Order and No Translational

Symmetry. Phys. Rev. Lett. 1984; 53(20): 1951-1954.

Srinivasan G, James CM, Krzycki JA. Pyrrolysine Encoded by UAG in Archea: Charging of a UAG-Decoding Specialized tRNA. Science 2002; 296: 1459-1462.

Trifonov EN, Bettecken T. Sequence Fossils, Triplet Expansion, and Reconstruction of Earliest Codons. Gene 1997; 205: 1-6.

Trifonov EN. Evolution of the Code and the Earliest Proteins. Reconstruction from Present-Day Sequences. Biophysics 2002; 47: 581-586.

Trifonov E.N. Concensus Temporal order of Amino Acids and Evolution of the Triplet Code. Gene 2000; 261: 139-151.

Trifonov EN. Presentation.


Trifonov EN. The Triplet Code From First Principles. J. Biomol. Struct. Dyn. 2004; 22(1): 1-11.

Vaesi A, Barkeshli M. Fibonacci Anyons From Abelian Bilayer Quantum Hall States. Phys. Rev. Lett. 2014; 113 (23):236804. (arxiv:1403.3383[cond-mat.str-el])

Wille JJ. Occurrence of Fibonacci Numbers in Development and Structure of Animal Forms: Phylogenetic Observations and Epigenetic Significance. Natural Science 2012; 4(4): 216-232.

Woese CR. On the Origin of the Genetic. Procs. Natl Acad Sci USA 1965; 54: 1546-1552.

Wong JT. A Co-Evolution Theory of the Genetic Code. Procs. Natl Acad Sci USA 1965; 72(5): 1909-1912.

Yang CM. The Bi-Pyramidal Nature, The Lucas Series in the Genetic Code and their Relation to Aminoacyl-tRNA Synthetases. 2003 (arxiv:q-bio/0309008)

Zhang A, Gladyshev VN. High Content of Proteins Containing 21st and 22nd Amino acids, Selenocysteine and Pyrrolysine, in a Symbiotic Deltaproteobacterium of Gutless Worm Olavius Algarvensis. Nucleic Acids Research 2007; 35(15): 4952-4963.

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The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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