# Fibonacci meets DNA: Molecular Mathematics

Fibonacci’s Golden Ratio has been applied to everything from honeybee populations, to conch shells and Tom Cruise’s face. However, the endless proportionality doesn’t stop there. It seems that the trippy number sequence reappears again in all things natural – including the molecular backbone of life.

The Fibonacci sequence is infinite, defined as the sum of two adjacent terms, where the first two terms F(0) and F(1) are 1 and 1, respectively.
Formulaically, this is: F(i) = F(i-2) + F(i-1).

Some background on the genetic code
The DNA code itself possesses certain invariable numerics. There are 4 nitrogenous bases that make up the genetic sequence of DNA: G, C, A and T (U in RNA). These are read in sets of threes (“codons”) when translated into a protein. Table 1: Amino acid codons. All proteins commence with Met (methionine). Stop codons terminate protein translation.

The central dogma of genetics states that to make a protein, DNA must be transcribed to messenger RNA and translated into a protein.

Therefore:
DNA:           ATG GAA  TTT  TGT TAA     becomes
mRNA:       AUG GAA UUU UGU UAA    becomes
protein:     Met   Glu  Phe   Cys

Consequently, as there are only 20 amino acids, but 43 = 64 codons, there is a degree of degeneracy to the molecular code (Table 1), such that a single amino acid may be encoded by more than one triplet base sequence.

A formula for the genetic code
Mathematicians have been trying to describe the formulaic basis of the genetic code for generations. The work is frequently collaborative and refers back to prior definitions of the mathematical similarities (atomic mass, redundancies, number of neurons) between different classes of molecules.

Négadi (2014) showed that by summing the first six Fibonacci terms, it is possible to obtain the number of amino acids (1+1+2+3+5+8=20). Table 1 indicates that there is one amino acid encoded by three codons, two amino acids encoded by one codon, three amino acids encoded by six codons and five amino acids encoded by four separate codons.

Using the class definition as prescribed by Rakovevic (Table 2), who allocated amino acids based to classes using the atomic numbers of their carriers (aminoacyl-tRNA synthetases), it is possible to describe the number of amino acids in terms of their class. Table 2: Classes of amino-acid carrier molecules (tRNAs) as described by Rakovevic.

These classes can be described by their Fibonacci terms, where plugging in the class number pops out the value of redundant codons:
F(1) = 2      Met; Trp
F(2) = 3      Ile
F(3) = 3      Leu, Ser, Arg
F(4) = 5      Phe; Ala; Thr; Val; Gly.
It has been proposed that if one sums the remaining terms, 1 (F(1)) and 8 (F(6)), it produces the number of amino acids encoded by two codons.
A formulae proposed by Gavaudin in 1971 also accounted for the number of even amino acid classes:
2 codons = 23 + 1
4 codons = 22 + 1
6 codons = 2 + 1
Here, the nine amino acids with two codons are described by 23 + 1, which correlates with the Fibonacci-derived result, 8 + 1.
Table 3 indicates how this may be resolved into the number of degenerate codons. Table 3: Summary of degenerate codons as determined by Négadi (2014)

Only a small taste into the extensive work in this field has been described above. For the full publication, click here.

Golden ratios in the helix
Rosalind Franklin first imaged the DNA molecule by X-ray crystallography, indicating it to be a double helix (“twisted ladder”). Since then, the proportions of different DNA conformations have been described, as the molecule adopts a more compact structure when it is silent (not being expressed), and a looser one when proteins require access to the major groove. Moreover, like all molecules, DNA changes its structure in water. The three conformations of DNA are characterised by different structural proportions.

B DNA is the most common conformation of DNA, and is 21 Å (angstrom; 0,1 nm) wide. It takes 34 Å to complete a single twist.

How phi presents itself in the backbone
If one expands the Fibonacci sequence, it reads: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55… and so on. Forever.
If one take any of the terms and divide it by its preceding term, the result is a non-terminating, non-precurring decimal. This value (1,61…) is phi, the Golden Ratio.

The great Ah-Whoom is the interplay between the distance and width of the B DNA backbone: phi. It has also been reported that the decagonal structure of the DNA molecule when viewed top-down, the way Franklin must have seen it, further possesses a phi ratio.

Tricksy proportionalities
Some consider the Golden Ratio proof of the underpinning perfection of nature, possibly even Intelligent Design. The verdict is out as to whether the re-emergence of these values in the genetic structure is a contrived extrapolation from coincidence, or evidence of the evolutionary functional of biological function. However, it is somehow comforting that at its most basic, the unassuming DNA backbone can be tied to the same commonplace miracles that emerge when the natural world is inspected closely enough.

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