In this paper, we present a new way to study the genetic code and its symmetries, using mathematics, more precisely, a small set of brand new Fibonacci-like sequences and, occasionally, some (useful) well known elementary functions from number theory. Using these sequences, the whole and detailed chemical content of the set of amino acids, as structured by several well known symmetry patterns, including their degeneracy, is revealed. Also, several other original applications, using the above sequences, are carried out. Before presenting all these results, let us begin by giving a brief summary of the genetic code. This latter is the basis of life on Earth and was masterfully deciphered in the mid-sixties of the last century [1]. It is the great biological “dictionary” that translates the language of DNA/RNA, which transmit the inherited information located in the genes, to the language of proteins which carry out the biological constructions and functions. It is well known that the “alphabet” of the former language consists of four fundamental units, the nitrogenous bases T (thymine), C (cytosine), A (adenine) and G (guanine) for DNA and U (uracil), C, A and G for RNA. As for the “alphabet” of the second language, it comprises the set of 20 amino acids. In the process of translation between these two languages, in the ribosome for short, there are ${64=4}^3$ “words”, the codons, which are made of three nucleotides. In the standard genetic code, only 61 of such codons are translated into amino acids and three others serve as termination or stop codons. To a given codon, from the set of 61, corresponds one and only one amino acid. Now, it is well known that several codons could code for the same amino acid so that the genetic code is said degenerate and the 20 amino acids are organized into “multiplets”. These are as follows: 5 quartets, each, coded by four codons proline (P, Pro), alanine (A, Ala), threonine (T, Thr), valine (V, Val) and glycine (G, Gly); 9 doublets each coded by two codons phenylalanine (F, Phe), tyrosine (Y, Tyr), cysteine (C, Cys), histidine (H, His), glutamine (Q, Gln), glutamic acid (E, Glu) and aspartic acid (D, Asp), asparagines (N, Asn) and lysine (K, Lys); 3 sextets each coded by 6 codons serine (S, Ser), arginine (R, Arg) and leucine (L, Leu); 1 triplet coded by three codons isoleucine (I, Ile) and finally 2 singlets each coded by one codon methionine (M, Met) and tryptophane (W, Trp). In the parentheses, the one-letter and three-letter codes of the amino acids are indicated. Below, in Table 1, we show the genetic code table (the three stop codons are indicated in green color background).

In this work, the “anomalous” three amino acids serine, arginine and leucine, each coded by six codons, will play a prominent role. Contrary to all other 17 amino acids the codons of which share the same first base, the three mentioned amino acids have, each, their 6 codons distributed into two different family boxes. (There are 16 such family boxes and each one of them is a set of four codons having the same first and second base (see Table 1)). The structure of the three sextets is the following serine: {UCN, AGY}, arginine {CGN, AGR}, leucine {CUN, UUR} (N for any base, Y for pyrimidine U or C and R for purine A or G). There are more and more voices rising to underline or put emphasis on the singular nature of the three sextets and also to bring experimental data which tend to show it [3,4]. On the other side, people have always taken for granted that the number of amino acids encoded by the (standard) genetic code is 20. Yes, this is true but this view is evolving. Some few years ago, a published work, [5], claimed that this number has to be increased to 23 by considering that the quartet part and the doublet part of each one of the 3 sextets as distinct. In this work, the authors present a new “effective number of codons”, called ${\mathrm{N}}_{\mathrm{c}}$, to characterize codon usage bias in the analyzes of protein-coding genes, which improves existing ones. Here, ${\mathrm{N}}_{\mathrm{c}}$ is shown to be a better predictor when its value is increased from 20 to 23 with, in particular, each 6-fold codon set (each sextet as it is called in this work) is considered to be composed of separate 4-fold and 2-fold parts. These six entities are ${\mathrm{S}\mathrm{e}\mathrm{r}}^{\mathrm{I}\mathrm{I},\mathrm{I}\mathrm{V}}, {\mathrm{A}\mathrm{r}\mathrm{g}}^{\mathrm{I}\mathrm{I},\mathrm{I}\mathrm{V}}, {\mathrm{L}\mathrm{e}\mathrm{u}}^{\mathrm{I}\mathrm{I},\mathrm{I}\mathrm{V}}$ which, added to the 17 remaining amino acids with no “degeneracy” at the first base position, as mentioned above, give a total of 23. This number, together with the remaining degenerate codons, 38, constitutes the pattern “$23+38$” the importance of which not only has been established in our recent works (see [2] and the references therein) but is also again established all along the present work. One of these kind of approaches is particularly relevant to the present work, the “Ideal” symmetry classification scheme, introduced some few years ago in [6]. It will be summarized in Section 2.3 and we present its numerous links with the present work in Section 4.2. In Section 2, we summarize three important symmetries of the genetic code, including the above mentioned one. In Section 3, we present our new Fibonacci-like sequences and their properties, which are the main mathematical tools used in this paper. In Section 4, we apply these sequences to derive the degeneracy structure of the 61 amino acids, their hydrogen atom, atom and nucleon number content, as structured by the symmetries mentioned in Section 2 and various other remarkable patterns. In Section 5, still using some elements of our sequences, we make contact with the work by shCherbak, [7], concerning the singular structure of proline and derive, in a totally original way, the mathematical form of the shCherbak-Makukov “activation key, [8], which, as is well known, lead to many remarkable and beautiful nucleon number patterns comprising, in particular, those related to Rumer’s symmetry (see below about this latter). In Section 6, using the “seeds” of our Fibonacci-like sequences, that is their “initial conditions”, and only these, we find that they are capable, on their own, to provide the very hydrogen atom content of the 61 amino acids, derived in the various patterns considered in Section 4. We strongly recommend the reader, at this point, before going to the next sections and get a comfortable reading of them, to take a look at Appendix A, which summarizes the chemical data of all 20 amino acids, including also the degeneracies and Appendix B, where a few other mathematical tools, used in this paper, are defined with the visualization of some computation examples.

Let us introduce now, as stated in the introduction, four Fibonacci-like sequences which will prove resourse-rich and very interesting in their applications throughout this work. (Another fifth sequence, just as interesting, will be introduced later, in Equ.(26).) They are also called $\left(\mathrm{p},\mathrm{q}\right)$-Fibonacci sequences and are defined by the following common defining relation where ${\mathrm{F}}_{\mathrm{n}}$ is the ordinary Fibonacci number. These four sequences differ only by the data of the numbers p and q which, here, play the role of “initial conditions” or “seeds”, as we will call them throughout this paper. Below, in the next section, we shall explain and justify the choice of these “seeds” but, for the moment, we introduce the four mentioned sequences by giving a name to each one of them and their “seeds”: (i) ${\mathrm{a}}_{\mathrm{n}}:\mathrm{p}=1,\mathrm{q}=6$, (ii) $a_n^{\text{'}}:\mathrm{p}=6,\mathrm{q}=1$, (iii) ${\mathrm{b}}_{\mathrm{n}}:\mathrm{p}=9,\mathrm{q}=13$, (iv) ${\mathrm{c}}_{\mathrm{n}}:\mathrm{p}=5,\mathrm{q}=30$. In Table 4 below, we give the first few terms

These sequences obey several linear relations, some of which will prove very interesting in view of their applications in this work. They are presented below, in Equ.(2), and could be easily checked

It is interesting to note here that the difference gives the (slightly modified) Fibonacci sequence noted ${\mathbf{F}}_{\mathbf{n}}^{\mathbf{\text{'}}}$ in an unusual but interesting form: its “seeds” here are inverted with respect to the usual Fibonacci sequence. Also, the sum of any of its first members until a certain index gives a Fibonacci number, exactly, contrary to the usual Fibonacci sequence with seeds 0, 1 which always gives one unit less than a Fibonacci number. For example, in our case, for $\mathit{n}=9$, we get ${\sum }_1^9{\mathbf{F}}_{\mathbf{n}}^{\mathbf{\text{'}}}=34$. (Note that the indexing is here shifted but the recurrence relation is still valid.) There is also another relation linking the sequences ${\mathbf{a}}_{\mathbf{n}}^{\mathbf{\text{'}}}$ and ${\mathbf{b}}_{\mathbf{n}}$. It writes

For $n=7$, the sequences $a^{\text{'}}$and $\mathrm{b}$ take the same value: ${\mathrm{a}}_7^{\mathrm{\text{'}}}-{\mathrm{a}}_5=0$. Also, for $n=8$, ${\mathrm{a}}_8^{\mathrm{\text{'}}}=86$ and $b_6=84$ and their difference is 2. The case $\mathrm{n}=9$ is also interesting, see below. These relations will have applications in the following sections. Importantly, the sequences in Table 4 together with the one defined in Equ.(26), below, display several numbers highly relevant in this work, either directly as members in Table 4 (shown in bold) or as sums to be evaluated in the following sections. We have also discovered that the above sequences, including the one defined in Equ.(26), can all be shown to exhibit a striking bilateral symmetry and other symmetry properties, in the line of thought of those established for the Fibonacci sequence by Edge, [11]. These findings will be reported elsewhere, in a forthcoming publication.

In this section, we use our Fibonacci-like sequences to shed light, by giving concrete results, on a question relative to the special amino (more exactly imino) acid, proline, which is an exception among the set of 20 amino acids. As a matter of fact, it is the only amino acid where its side chain is connected to its block twice. shCherbak, [7], in order to “standardize” the common block of the amino acids, with 74 nucleons, proposed an imaginary “borrowing” of one nucleon (in fact one hydrogen atom) from the side chain of proline, which has only 73 nucleons in its block, to the benefit of its block, to reach 74, as the 19 other amino acids. In his next work with Makukov, [8], the above “borrowing”, or transfert of one nucleon, has been termed the “activation key”. Activating the key, i.e., standardizing, leads to an innumerable number of remarkable and beautiful arithmetical patterns. These authors say in their paper, [8], “Applied systematically without exceptions, the artificial transfer in proline enables holistic and arithmetically precise order in the code”. Here, in this section, we establish,, not only a mathematical version of the “activation key”, itself, but also its action on the (new) total hydrogen atom content, with simple possible extension to the atom and nucleon content. Let us begin by examining the action of the “activation key”). Consider, again, the sequence ${\mathrm{a}}_{\mathrm{n}}^{\mathrm{\text{'}}}$ and the following sum

It could be easily shown and verified that the above relation holds for any k. For k=9, it gives $358=364-6.$ As established and mentioned many times previously, $358$ is the number of hydrogen atoms in the 61 amino acids side chains, where the special amino acid proline has 5 hydrogen atoms in its side chain. If, instead, one considers that proline’s side chain has now 6 hydrogen atoms, at the cost of its block, i.e., no standardization made, or the “activation key” off (see below), and taking into account the number of its coding codons, which is 4, then we have now $362=358+4$ hydrogen atoms in the 61 amino acids side chains. Let us reconsider Equ.(10)) for the hydrogen atoms partition between 38 amino acids corresponding to the 38 degenerate codons $\left(219\right)$ and 23 amino acids ($139$) but, now, using the above relation ($358=364-6$):

To get a correct partition, let us consider the perfect number 6 which is, as such, equal to the sum of its proper divisors: $6=1+2+3$. These are just the right numbers we need. Inserting these in the above equation by selecting the odd divisors $1$ and 3 and shifting them to the left while leaving the even one, 2, to the right and finally arranging properly, we get

so that we have something quite correct: one more hydrogen atom in the 23 amino acids part (proline, a quartet, has now one more hydrogen atom in its side chain) and 3 more hydrogen atoms for its 3 degenerate codons. (Note that, the action of the “activation key” could easily be extended to the total number of atoms and the total number of nucleons, as it applies only to the hydrogen atom number content.) Taking a look at the 6^th term in the sequence ${\mathrm{c}}_{\mathrm{n}}$, $115=40+75$, we have that it appears to be equal to the number of nucleons in proline side chain and block, see below about this latter sum. This number, 115, is invariant whether we make shCherbak’s “borrowing” of one nucleon or not. To get more insight, we consider another invariant number, the total number of hydrogen atoms in the 61 amino acids, including the blocks (with 4 hydrogen atoms in each), that is $358+244=362+240=602$. Without borrowing one nucleon from the side chain of proline in favour of its block there are 362 hydrogen atoms in the 61 amino acids and 240 hydrogen atoms, $57\times 4+4\times 3=240$, in the 61 blocks. Applying the “borrowing”, there are 358 hydrogen atoms in the 61 amino acids side chains and $244(=61\times 4)$ hydrogen atoms in the 61 blocks. Note, in passing, the following nice relations seemingly linking the two views: $\left(240\right)+\left(362\right)=244$ and $\left(240+362\right)-\left[\left(240\right)+\left(362\right)\right]=358$. Now, let us examine the former point,, the derivation of the “activation key”. Considering the above mentioned invariant numbers, $115$ ($=5\times 23)$) and $602 \left(=2\times 7\times 43\right)$, we have, using their ${\mathrm{A}}_0$ function (see Appendix B):

From which we deduce that $115=42+73=41+74$ which are seen to describe, fully and precisely, the two views: $42+73$ (“activation key” off) and $41+74$ (“activation key” on). From $\left(41\right)=42=41+1$, where

is the sum of the divisors, we can also write $115=\left(41+1\right)+73=41+(73+1)$. Also, from $\left(41\right)=40=41-1$, we can also make contact with the sequence ${\mathrm{c}}_{\mathrm{n}}$ through the relation $c_6=115=40+75$, mentioned above: $41+\left(75-1\right)=41+74$. Moreover, we can, alternatively, exploit the number 75, itself. As a matter of fact, calling Legendre’s sum of three squares theorem[3], we have that the number 75 do not satisfy the theorem and can therefore be written as the sum of the following three squares: $1^2+5^2+7^2$ or $1+\left(25+49\right)=1+74$. This latter form gives us again $\left(40+1\right)+74=41+74$. Finally, using $\left(41\right)=40=41-1$ and the decomposition of the number $75$ as the sum of three squares, mentioned above, we can write, by allocating the two units in two ways: $41-1+1+74=41+74=42+73$. This is, again, what we found above from Equs.(52-53).

We invite the reader, here, before starting this section, to remember what has been said about the three sextets in Section 4.2 when using the “seeds” $6 \mathrm{a}\mathrm{n}\mathrm{d} 1$ of our fifth sequence ${\mathbf{g}}_{\mathbf{n}}$. In the “ideal” symmetry classification scheme, mentioned in Section 2.3, the authors explain that, in their approach, the symmetries are a consequence of sextet’s dynamics and the whole set of amino acids is created starting from these three sextets where, serine, plays a prominent role. In our own approach, relying on the use of Fibonacci-like series, on the other hand, we have chosen, as already mentioned, for two of them, ${\mathrm{b}}_{\mathrm{n}}$ and ${\mathrm{c}}_{\mathrm{n}}$, the hydrogen atom numbers and atom numbers of the three sextets (see Section 4.2) as “seeds”. It appears, while the writing of this paper goes to its end, that the “seeds” of, in fact, all the Fibonacci-like sequences used in this paper, and only these, by themselves, can “create”, strikingly and remarkably, the main hydrogen number patterns derived in this paper. As a matter of fact, the sum and product of the “seeds” of the sequence ${\mathrm{b}}_{\mathrm{n}}$, alone, gives

One recognize here the number of hydrogen atoms in 20 amino acids, 117, augmented by the number of hydrogen atoms in the three sextets, 22. The total, 139, corresponds to 23 amino acids (the sextets counted two times). Now, let us compute the following expression, using the sum and product of the “seeds” of the sequence ${\mathbf{c}}_{\mathbf{n}}$ and only the sum of the “seeds” of the other three remaining sequences ${\mathrm{a}}_{\mathrm{n}},$ ${\mathrm{a}}_{\mathrm{n}}^{\mathrm{\text{'}}}$, and ${\mathrm{g}}_{\mathrm{n}}$ (the latter defined in Equ.(26). We have

Here, we have the number of hydrogen atoms in the 38 amino acids corresponding to the 38 degenerate codons. Equs.(54-55), together, constitute the “$23+38$” hydrogen atom pattern established in Section 4.1). Now, taking the sum of the above equations and borrowing the number 22 from Equ.(54) to the benefit of Equ.(55) gives $117+241=358$ which corresponds to the other pattern “$20+41$” (20 amino acids and 41 degenerate codons), see Equ.(11) in Section 4.1. Next, we arrange Equs.(54) and (55) as follows

Here, we have again the hydrogen atom content in Rumer’s division: 172 hydrogen atoms in ${\mathrm{M}}_1$ and 186 hydrogen atoms in ${\mathrm{M}}_2$, see Section 4.2 Equ.(20). To get the other patterns, we call, exceptionally, the Fibonacci ($0, 1, 1, 2, 3, 5, \dots$) and Lucas ($2, 1, 3, 4, 7, 11, \dots$) series, which, as well known, are linked by the relation ${\mathrm{F}}_{\mathrm{n}}+{\mathrm{L}}_{\mathrm{n}+2}={\mathrm{F}}_{\mathrm{n}+4}$. For n=5, we have $5+29=34$ so that we can take this latter as the term $14+20=34$ in the above equations, arrange and get

This is the hydrogen atom pattern for (i) the 3^rd base classification of Section 4.2, Equ.(14), and (ii) the “ideal” symmetry classification scheme in the same section, Equ. (22). Finally, we consider Zekendorf’s theorem (see footnote 1) and apply it to the number 117 giving $89+21+5+2$. Writing $89$, a Fibonacci number, as $55+34$, we can rearrange the content of the second parenthesis in Equ.(57) above as $55+29=84$ and $34+21+22+5+2=84$, so that $168=2\times 84,$ which describes again the pattern $190+2\times 84=358$. The fact of having used, here, the Fibonacci and Lucas sequences is all the more interesting in that it can, also, give us another remarkable result. As a matter of fact, adding the two “seeds” of the Fibonacci and Lucas sequences, $0 \mathrm{a}\mathrm{n}\mathrm{d} 1$ and $2 \mathrm{a}\mathrm{n}\mathrm{d} 1$, respectively, to the above sum of Equs.(54) and (55) and arranging, we obtain

This is, exactly, the hydrogen atom pattern found in Section 5, devoted to the special imino acid proline and the shCherbak-Makukov “activation” key, when this latter is “off”, see Equ.(51) and below it, in Section 5.

In this work, we have strayed a little off the beaten paths in the genetic code mathematical research. Starting with a handful of Fibonacci-like sequences, we have derived not only the degeneracy structure of the genetic code but also the hydrogen atom content, the atom number content and also the integer molecular mass (nucleon) content of the set of 20 amino acids as strutured in the 64-codon table. What made this possible is a judicious choice of the intial conditions, or “seeds”, of the above mentioned sequences. For two of these, they are chosen as the hydrogen atom numbers and atom numbers of the three enigmatic and intriguing amino acids serine, arginine and leucine. Our results, using these sequences, led us to reveal, as mentioned above, the elemental content of the 61 amino acids set as structured by various well-known symmetries, as Rumer’s symmetry, the “ideal” symmetry classification scheme and the basic symmetry associated to order-2 degeneracy. Moreover, as a by-product of our mathematical formalism, we derived the atomic (elemental) content of the building-blocks of RNA, the four ribonucleotides UMP, CMP, AMP and GMP. Also, still using the above mathematics, we bring, for the first time, an additional brick to shCherbak’s theory concerning the role of the special imino acid proline whose virtual “double” struture renders possible, via the use of the “activation key”, a large number of remarkable and beautiful arithmetical patterns. Let us stress, as a last word, that the content of this paper is, we believe, inovative. To the best of our knowledge, we have never seen such a kind of quantitative derivation of the chemical characteristics of the genetic code as structured by the degeneracy and also by several well known symmetries. Our main findings, as the total hydrogen atom content, the total atom content, the total molecular mass content of the 20 amino acids, including the degeneracy, as well as other relevant quantities related to the symmetries of the genetic code are found directly, either as ostensible members of the Fibonacci-like sequences or obtained from the summation properties of these latter. Other slighly more sophisticated quantities, also having something to do with the symmetries, are obtained with the help of some well known arithmetic functions.