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/" %0 1 / 2-*3 & # " ' ! ( ' ) # # 2-*3 Miloje M. Rako64
G E N E T I C C O D E The Cipher and the Key
LAMBERT Academic Publishing
Contents
Chapter 1 The Cipher and the Key
1. INTRODUCTION ...... 1 2. A POSSIBLE SCENARIO OF SELECTION OF CONSTITUENTS ...... 3 O P ...... 3 2.2. The starting molecules and derivatives ...... 11 2.2.1. Derivative of derivative ...... 11 2.2.2. Diversity enriching and economicity ...... 13 2.3. The branching structures ...... 14 2.3.1. The "mapping" from the head to the body ...... 15 2.3.2. The pure hydrocarbon AAs ...... 15 3. NUCLEOTIDE TABLES AND THEIR FACES ...... 16 3.1. The 24 permutations of four Py-Pu bases ...... 16 3.2. The nucleon number balances ...... 16 3.3. The atom number balances ...... 23 3.4. The sense of the modification of R ...... 25 4. A SPECIFIC CHEMICAL COMPLEMENTARITY AS THE CIPHER ...... 27 4.1. Three types of distinct chemical complementarity ...... 28 4.2. Nuancing and balancing of chemical structures and properties ...... 29 4.3. The chemical complementarity as a neighborhood logic ...... 30 4.3.1. The best possible permutation arrangement ...... 30 4.3.2. Dissimilarity in similarity ...... 32 4.4. The cipher in relation to particles balance ...... 33 5. THE POLARITY IN RELATION TO THE CIPHER KEY ...... 34 5.1. The final result ...... 34 5.2. The final result in relation to some previous ones ...... 39 6. CONCLUDING REMARKS ...... 40
Chapter 2 Analogies of Genetic and Chemical Code...... 45 1. Preliminaries ...... 45 1.1. Analogies with quantum physics ...... 45 1.2. Agreement-disagreement principle ...... 47 1.3. Some other mathematical formalisms ...... 48 2. New insights ...... 49 2.1. The problem of lanthanides arrangement in the PSE ...... 49 2.2. Specific arithmetical patterns ...... 50 2.3. Quantum like relationships within nucleotides arrangements ...... 58 2.4. Specific algebraic patterns ...... 63 2.5. Key relationhips between GC and PSE ...... 63 2.6. Some additional observations on Genetic code ...... 65 3. Discussion ...... 68 3.1. Another understanding of the periodic system ...... 69 3.2. Another similarities ...... 70 3.3. Some additional observations ...... 70 4. Conclusion ...... 71
III Chapter 3 Chemical Distinctions of Protein Amino Acids within Genetic code ...... 83 REFERENCES ...... 97
Chapter 4 The Enigma of Darwin's Diagram ...... 101 1.Introduction ...... 102 2. Methodology ...... 103 4. Results and discussion ...... 106 4.1. Primary and secondary branches of species "A" and "I" ...... 106 4.2. The riddle of the genetic code ...... 108 4.3. Darwin's solution to the riddle of the genetic code ...... 108 4.4. Nodes and branchings ...... 110 4.5. BinaryMcodeM "#R " ...... 111 4.6. The balances of the number of branches for two species ("A" and "I") ...... 111 4.7. The "Prime Quantum 037" ...... 113 4.8. Primary and secondary branches for "other nine species" ...... 113 4.9. Primary and secondary branches for all 11 species ...... 114 4.10. Improbable and unexpected result ...... 115 4.11. More than improbable result ...... 116 5. Concluding remarks ...... 116 REFERENCES ...... 196
Supplement 1 Genetic Code as a Coherent System ...... 201 1. Preliminaries ...... 201 2. Four stereochemical types of AAs ...... 205 3. Four diversity types of AAs ...... 209 3.1. The uniqueness of the number 26 ...... 210 3.2. The unity of two realities M chemical and mathematical ...... 212 3.3. The uniqueness of the number 48 ...... 216 4. Harmonic structures ...... 219 4.1. Determination by Golden mean ...... 219 4.2. Golden mean M enzyme determination ...... 222 4.3. The Li $ R ...... 224 4.4. A specific harmonic structure ...... 228 5. Concluding remark ...... 230 Appendix: Some source harmonic structures ...... 231
Supplement 2 Golden and Harmonic Mean in the Genetic Code ...... 236 1. Introduction ...... 236 2. A new rearrangement of nucleotide doublet Table ...... 237 3. Particles number balances through polarity ...... 238 4. Determination through "golden whole" ...... 239 5. Concluding remark ...... 240 REFERENCES ...... 241
IV V FOREWORD
Exactly sixty years ago, when I started chemistry and biology studies, I have been occupied by the idea that there must be an essential reason why the key building chemical elements of living matter are those which are, as the lightest non-metals ones, on the "top" of the Periodic system of chemical elements (PSE): hydrogen, carbon, nitrogen, and oxygen. Thirty years later, I published the book "Genes, molecules, language" (in Serbian with an English language supplement) in which there was still no answer to the question asked. For the most parts, the book concerned genetic code (GC) and contained some of my research results, including the binary tree of the genetic code. In a way it seemed that I gave up searching for a chemical answer to the question asked, and latched on the mathematical-information tools, what the trend in the GC research has already been largely revived. Why I would not do so, when it was directly obvious, to other researchers and to me, that the state of things in the Table of GC is such that each of the 64 codons could encode any of the 20 protein (canonical) amino acids. Together with this chemical, one biological apparition was imposed: it cannot be anything other than Darwin himself, with his selective sieve, "included his fingers". Of the many possible molecular aggregations in the prebiotic "soup," Nature chose the very one who produced this chemical state, which proved to be the only capable of producing the totality of life we know. But, the "worm of suspicion" did not abandon me. It must be that some hidden chemical lawmaking hides in all of this. I continued my research and published the results, and there were many of them, what is also indicated by the fact that after the aforementioned 1988 book, I published three other books on genetic code, one in Serbian and two in English, plus several scientific papers in national and international journals. No, there was no response in them to the asked question. Thirty years more have passed and the answer is, finally, here. More accidentally, I deliberately looked back at the GC binary tree again. With surprise I viewed that the beginning of the binary tree, with a purely chemistry "thirst," seeks to connect with the end; a step behind the beginning with a step ahead of the end, and so on M the chemical zipper was made. The four chapters of this book with two supplements testify that. In the first chapter is presented a new approach to understanding of the genetic code. In order to overcome the key paradox (and Darwinian selection problem) that the highly complex amino acid phenylalanine is encoded by the simplest codons (UUY), and the simplest glycine encoded by the most complex codons (GGN) as well as the paradox of the duplication of some amino acids in the genetic code table (leucine, serine and arginine), we proposed an extension of the notion (and concept) of genetic code. For a better understanding of genetic coding, we proposed a hypothesis after that genetic code has to be understood, analogously to understanding in cryptology, as the unity of the three entities: the code, the cipher of the code and the key of the cipher. Only in the first chapter we are talking directly about the concept of the cipher, while in the remaining three chapters and in two supplements it is implied. Moreover, the contents of these chapters themselves provide a "cipher" in a wider context; it is
VI understood that the genetic code is hierarchically encrypted. For these reasons, the concept of code had to be extended in a different way. Hence, it makes sense to note that the second chapter discusses the analogy between genetic and chemical code, under which the chemical code does not include the periodic table of chemical elements such as "Table", but as a mathematical expression, analogous to the mathematical expressions valid for the genetic code of which is the most important one according to which the generation of codons corresponds to the generating the third-class variations with repetition from a set of four elements. Between others relations, it appears also a correspondence between the distribution of codons in the genetic code table and the distribution of chemical elements in the periodic system with respect to their even/odd parity and stability/instability of their isotopes. In the third chapter it is shown that twenty protein amino acids appear to be a whole and very symmetrical system, in many ways, all based on strict chemical distinctions from the aspect of their similarity, complexity, stereochemical and diversity types. By this, all distinctions are accompanied by specific arithmetical and algebraic regularities, including the existence of amino acid ordinal numbers from 1 to 20. Also, the classification of amino acids into two decades (1-10 and 11-20) appears to be in a strict correspondence with the atom number balances. In the fourth chapter for the first time is presented a hypothesis, that the one and only "accompanying diagram" in Darwin's famous book on the origin of species contains, may be, a hidden code. Starting from the basic structure of the Diagram, Darwin formed a sophisticated structure which strictly corresponds to the arithmetical and /or algebraic structures that also appear to be the key determinants of the genetic code. From these results it follows that Darwin with his Diagram anticipated the relationships not only in terrestrial code but in the genetic code as well, anywhere in the universe, under conditions of the presence of water, ammonia and methane, phosphine and hydrogen sulfide. If so, then Darwinian selection moves one step backwards in prebiotic conditions, where it refers to the choice of the life itself. Over the course of forty years of my researches of the genetic code, there were many people who, in various ways, helped me to pursue this path. To most of them I thanked in my previous books and papers. I want to thank them on this occasion even though I do not name them. To those who are helping me, and they helped me in preparing of this book with their stimulating discussions and benevolent critic, I am especially grateful, % & 64' () " , Djuro Koruga, D& )4' Tidjani Négadi' & *)4, Lidija Matija + -4 I am very grateful to the Publishing House Lambert Academic Publishing for noticing my research and proposing to publish this book. Special thanks to the editor Lucia Jardan, who exerted oneself a grait lot of effort and patience and gave me great help to keep the book up to the desk . / *& 4 0&4 " 0&4 -preparation of the manuscript for the press.
Belgrade, 18.04.2018. 64
VII Chapter 1
The Cipher and the Key
In this chapter is presented a new approach to understanding of the genetic code. In order to overcome the key paradox (and Darwinian selection problem) that the highly complex amino acid Phe is encoded by the simplest codons (UUY), and the simplest Gly encoded by the most complex codons (GGN); as well as the paradox of the duplication of some amino acids in the encoding process (Leu, Ser, Arg), we proposed an extension of the notion (and concept) of genetic code. For a better (and lighter) understanding of genetic coding, we proposed a hypothesis after that ^%nder the conditions of allowed metaphoricity and modeling in biology) genetic code has to be understood, analogously to understanding in cryptology, as the unity of the three entities: the code, the cipher of the code and the key of the cipher. In this hierarchy the term (and notion) "genetic code" remains what has been from the beginning: a connection between four- letter alphabet (four Py-Pu nucleotides, in form of codons) and a twenty-letter alphabet (twenty amino acids); the cipher is a specific chemical complementarity in chemical properties of molecules in the form: similarity in dissimilarity versus dissimilarity in similarity ("Sim in Diss vs Diss in Sim") and the key of cipher: the complementarity on the binary tree of the genetic code in the form: 0-15, 1-14, 2-13, ..., 6-9, 7-8. Just only with this understanding, it appears a possibility for an additional understanding that within the two main Genetic Code Tables (of the nucleotide doublets and nucleotide Triplets) exists a sophisticated nuancing and balancing in the properties of th constituents of GC, including the balance of the number of molecules, atoms, and nucleons.
1. INTRODUCTION
From the time when the genetic code (GC) began to be considered practically deciphered (Crick, 1966; Rumer 1966), up to the present day, on the scientific scene is a paradigm according to which, in the interpretation of genetic code, the terms (and notions) "cipher" and "code" are synonyms. The word, however, must be about that thing, that in the interpretation of the genetic code, the terms "code", "cipher" and "key of the cipher" must be distinguished, analogously as it does in cryptology and cryptography. Just this position will be the backbone of this chapter. A little hint of such an idea, the idea of mediating by the "something" in the coding process, we had already have in the seventies of the last century. It was that T. Jukes has rebelled against the almost plebiscite attitude of the genetic code researchers that it exists a direct affinity between the codons and amino acids (AAs), and set out the hypothesis that it will be rather a kind of mediation, at least a mediation of specific adapters: "Various authors have advanced schemes for the origin of the genetic code based on proposals for direct affinity between amino acids and nucleotide bases or groups of bases. Such affinities are deemed t / Q RG The present genetic code, however, and presumably its immediate antecedents, involve the
1 intervention of adapter molecules ... which combine enzymatically with amino acids, rather than direct affinity between amino acids and nucleic acid molecules" (Jukes, 1973). True to the will, there were also other authors who suggested that it might be mediation in the affinity of codon and amino acids, even from the very beginning of understanding of genetic code. Thus, F. Crick (1966, 1968) could not understood (nor accepted) that in most cases similar codons (trinucleotide aggregations) encode similar AAs, and in three cases (Leu, Ser, Arg) similar and non-similar, start with the hypothesis, that, if coding does not hold affinity, i.e. stereochemical conditions, then encoding within the genetic code must be "mediated" by "pure chance". At the same time Y. Rumer (1966) suggests that encoding by dinucleotide aggregations is mediated by "grammatical" formalism (the relation between words and the root of the word), semantics (one-meaning and multy-meaning codon families) and by semiology, i.e. semiotics (the classification of nucleotide doublets1 after the number of their hydrogen bonds which apear here as "signifiant" and "signifié" (signifier and signified) t the same time, that is as their unity (De Saussure, 1985, p. 99). [In further examples for grammatical, semantic and semiological analogies, we will not cite the signs of the allegations, since these relationships in the science of genetic code are generally known.] Almost twenty years after the afore mentioned works by F. Crick and Y. Rumer in 1966, R. Swanson (1984) found that the coding in genetic code is mediated by Gray code based on the binary numbering system. Starting from R. Swanson's conclusion that "Gray code binary symbols [represent] the numbers 0-63", we have shown that the Gray code model as well as the Genetic Code Table (GCT) can be developed into binary tree with a codons display at exactly 0M63 3 )64' 455,b; 1998a).2 Ten years later, after the R. Swanson's work, V. Shcherbak showed that genetic coding is mediated by Pythagorean triples within specific patterns of the number of nucleons in canonical AAs, such patterns that they themselves "represent analogies with quantum physics" (Shcherbak, 1994). Now, in this chapter, it will be shown that this mediation is far wider. According to our hypothesis, the asignement of codons to amino acids is mediated by a specific cipher (and its key), based on a specific chemism in correspondence with the binary-code tree.
1 Nucleotides are the basic building blocks of DNA and RNA; the deoxynucleotides and ribonucleotides, respectively. As the monomers the nucleotides consist from the organic bases, the derivatives of Pyrimidine and Purine (Py-Pu) plus sugar (ribose or deoxyribose) and phosphate. For a general dessignation, we use for the nucleotides the abbreviated signifiers: Y for pyrimidine, R for purine and N for all four types of nucleotides. 2 The Genetic code six-bit binary tree represents, per se, the Boolean space, i.e. the Boolean cube Bn (n= 0, ' ' ' 6' 78 3 )64' 449' 44:8 From the chemical point of view, the cases for n = 2, 3, 4 are particularly important, where the latter case (n=4) represents a display of 16 nucleotide doublets.
2 [Nota bene: "Before discussing these problems ..., we must address a preliminary one. We must face the ontological problem of the reality of the organic codes: are they real codes? Do they actually exist in living systems? It is a fact that the genetic code has been universally accepted into Modern Biology, but let us not be naive about this: what has been accepted is the name of the genetic code, not its ontological reality. More precisely, the genetic code has been accepted under the assumption that its rules were determined by chemistry and do not have the arbitrariness that is essential in any real code. The theoretical premise of this assumption is the belief that there cannot be arbitrary rules in Nature, and this inevitably implies that the genetic code is a metaphorical entity, not a real code. This idea has a long history and let us not forget that for many decades it has been the dominant view in molecular biology" (Marcello Barbieri, 2018, p. 2).]3
2. A POSSIBLE SCENARIO OF SELECTION OF CONSTITUENTS 2.1. The OprinciplePof the genetic code With the revelation of the GCT, in both forms as the Table of the nucleotide triplets (Crick, 1966) and the Table of nucleotide doublets (Rumer, 1966) (Comment 1), it has become an obvious scenario by which the chemical constituents of the genetic code are generated; both builders, of nucleic acids, as well as of protein amino acids (Figures 1 and 2).
Comment 1. Both mentioned Tables (as Table 1.1 and Table 2.1, respectively) are given here, in a customized processing. Table 1.1 is a generalized GCT of nucleotide triplets in the sense that the triplets are reduced to doublets, according to the corresponding permutation of Py-Pu bases (Section 3.1), their positions in the Table of doublets (Table 1.2), as well as on the binary-coded tree (Rako64, 1998, Figure 1).
The first thing that is directly visible is the fact that all constituents (molecules) of genetic code are built from the first few simplest elements (non-metals) at the beginning of the Periodic System of Chemical Elements (PSE). From the first four elements (H, C, N, O), pyrimidine (Py) and purine (Pu), that is their genetic code derivatives, were constructed; also the 18 AAs. [These four elements are in immediate neighborhood, three in continuity (C, N, O), at the beginning of the 4th, 5th and 6th groups, respectively, and H at the beginning of the 7th group, as the diagonal neighbor of oxygen.] To complete the nucleotide molecules, and to build two more AAs, which have sulfur, the Darwin's selection sieve must expand its openings in order to select phosphorus (as the vertical neighborhood of nitrogen) and sulfur (as the vertical neighborhood of oxygen) from the next period of PSE.
3 OThe very first model of the genetic code was the Stereochemical Theory, an idea proposed by George Gamow in 1954 ... The second canonical model was the Coevolution Theory proposed by Wong (1975, 1981), according to which the genetic code coevolved with the biochemical pathways that introduced new a / P (Barbieri, 2018, p. 2) (The bolding: MMR). 3 From these facts it can be concluded that the structure of chemical constituents of genetic code, in part directly, and partly indirectly, corresponds to the "Aufbau principle" of PSE.
Table 1.1. The generalized nucleotide triplets Table
U C A G
C I UUN F II UCN UAN Y I UGN S II (0) L I (2) (8) ct (10) ct W I
CUN CCN CAN H II CGN L I P II R I 120 + 10 (1) (3) (9) Q I (11) (11)
AUN Ile I ACN AAN N II AGN S II T II (4) M I (6) (12) K II (14) R I
D II 119 -10 GUN GCN GAN GGN V I A II G II (5) (7) (13) (15) E I (12) (12)
(11-1) 119 - 20 (12+1) 120 + 20 53 77 46 63
The number of atoms within amino acid side chains corresponds to the number of atoms in the R ' / ;/' / changes are for ±10 and ±10 once more. The changes in molecules number for 1 and ±1. The relations to arrangements in Tables 2.1 & 2.2 as follows. The diagonal relation: 53 + 63 = 115+1 and 46 + 77 = 124- 6 < 9 R 3 8 6 < 9 R 3 8
Table 1.2. The nucleotide doublet Table with binary records
YMY (PyMPy) YMR (PyMPu) UU 0000 UC 0010 UA 1000 UG 1010 CU 0001 CC 0011 CA 1001 CG 1011 AU 0100 AC 0110 AA 1100 AG 1110 GU 0101 GC 0111 GA 1101 GG 1111 RMY (PuMPy) RMR (PuMPu)
4 Table 1.3. The Py-Pu and odd-even arrangement of amino acids within standard Genetic Code Table (GCT)
L1+ V2 + S3 + P4 + T5 + A6 + Y7 + R8 + G9 = 42 + 39 = 81
I FY + MR + HY + KR + DY + WR + SY = 37 + 44 = 81 LR + IY + QR + NY + ER + CY + RR = 51 + 26 = 77
II (FL)1+(MI)2+(HQ)3+(KN)4+(DE)5+(WC)6+(SR)7 =(37+51)+(44+26) = 88 + 70 III (42 + 88 = 120 + 10) (39 + 70 = 119 M 10) IV (88 M 77 = 11) (81 M 70 = 11)
The Table corresponds with Table 5.2. In both Tables are the same amino acid singlets, but here their order is according to the positions in GCT, and in Table 5.2 according to positions on the binary tree. Sector I: In the first row there are amino acid singlets ("lonely" AA in four-codon space), in the order as in GCT. In the second and third row are amino acid doublets (per two AAs within four-codon space). Sector II: The amino acid doublets. In Sector III it is shown that number of atoms in amino acid singlets and doublets is determined by the middle pair of the total number of atoms within the set of 23 AAs (0, 1, 2 , 3, 4, ..., 119-120, ..., 235, 236, 237, 238, 239), with a change for ± 10. In Sector IV the following ratio of the number of atoms was shown: the difference in the number of atoms in all doubled AAs at odd positions (88) and the number of atoms in purine and doubled AAs at even positions (77) is equal to the difference in the number of atoms in all single AAs (81) [or in all pyrimidine and doubled AAs (81)] and all purine and doubled AAs at even positions (70). The nuance balance is self-evident.
5 Table 2.1. The Rumer's nucleotide doublets Table (Rumer, 1966)
114 30 (119) 89 125 116 108 Gly GG (6) Phe UU (4) Leu Pro CC (6) Asn AA (4) Lys Arg CG (6) Ile AU (4) Met Ala GC (6) Tyr UA (4) ct
Thr AC (5) His CA (5) Gln Val GU (5) Cys UG (5) Trp Ser UC (5) Asp GA (5) Glu Leu CU (5) Ser AG (5) Arg
125 36 (120) 84 114 106 118 330-66 330±00
125 + 114 = 239 125 M 114 = 11
The one-meaning nucleotide doublets and corresponding four-codon amino acids on the left; and the two-meaning (UG as three-meaning) doublets and corresponding non-four-codon amino acids on the right. Four quartets (of nucleotide doublets) in relation to the number of hydrogen bonds. In the set of 23 AAs there are above 30 + 89 = 119 atoms in 11 amino acid molecules, within their side chains; and down: 36 + 84 = 120 atoms in 12 molecules; diagonally: 30 + 84 = 114 atoms in 12 molecules, and 36 + 89 = 125 atoms in 11 molecules. Within Py-Pu bases there are 106 + 116 atoms on the left and 108 + 118 on the right. [U = 12, C = 13, A = 15, G = 16; all as in (Rako64' 44:8] At the bottom (shaded) M the number of atoms in the amino acid molecules (side chains): within 32 amino acid molecules on the left and 29 on the right, within the set of "61" amino acid molecules. [Note: In originsl Rumer's Table only the number of hydrogen bonds (4 & 6 and 5 & 5 in brackets) is calculated; all other calculations are ours.]
6 Table 2.2 R = acid side chains, corresponding with odd/even positions
01. G GG (6) 02. F UU (4) 03. L ) 04. P CC (6) 05. N AA (4) 06. K 07. A GC (6) 08. Y UA (4) 09.ct 124 10. R CG (6) 11. I AU (4) 12. M 13. V GU 14. C UG 15. W (5) (5) ( ) / Even 16. T 17. H 18. Q
AC (5) CA (5) 115 19. L CU (5) 20. S AG (5) 21. R
22. S UC (5) 23. D GA (5) 24. E Odd ( 28 39 48 (10) 38 39 (00) 47 (01) 66 78 60+35
The odd positions of nucleotide doublets and corresponding AAs are shaded. On the odd/even positions there are 115/124 atoms, respectively, in a balance correspondence with the "diagonal result" in the original Rumer Table (125/114 and Table 2.1). The result within the columns (60, 66, 78 atoms within the amino acid side chain) corresponds with the number of atoms within 7 "golden" AAs, 7 their complements and 6 non- / 3 = )64' 1998a, Scheme 2, p 289; cf. Survey 3 in this chapter). A "surplus" of 35 atoms is a "balance fraction" which, when passing from 204 atoms in the set of 20 AAs to 239 atoms in the set of 23 AAs, corresponds to the quantity for three doubled AAs (L13 + S05 + R17 = 35).
7 Table 3. The Canonical Invariant Sistem (CIS) of codons and corresponding amino acids, according their positions on the binary-code tree (Rako64' 4458
0 UUN F, L 8 UAN Y 15 GGN G
1 CUN L 9 CAN H, Q 14 AGN S, R
2 UCN S 10 UGN C, W 13 GAN D, E
3 CCN P 11 CGN R 12 AAN N, K 53 77 63 46 63 77 4 AUN I, M 12 AAN N, K 11 CGN R
5 GUN V 13 GAN D, E 10 UGN C, W
6 ACN T 14 AGN S, R 9 CAN H, Q
7 GCN A 15 GGN G 8 UAN Y (53+46 = 119 - 20) (53+77 = 120+10) Left 46 56 Right (77+63 = 120 +20) (46+63 = 119 - 10) 53 84 46 + 56 = 102 (53+63 =115+1) (46 + 77 = 124-1) 53 + 84 = 102 + 35
The number of atoms in the upper and lower part of the Table corresponds to the number of atoms in GCT (Table 1.1). Diagonal result (115 + 1 & 124-1 vs 115 & 124 in Table 2.2). A "surplus" of 35 atoms is a "balance fraction" which, when passing from (102 + 102 = 204 atoms) in a set of 20 AAs to 239 atoms in a set of 23 AAs, corresponds to the quantity for three doubled AAs (L13 + S05 + R17 = 35).
8 Table 4. The vertical CIS display into one-meaning and two-meaning nucleotide doublets and corresponding amino acids
an on2 on1 c1 aa s aa c2 on1 on2 an
13 1 0001 CUN L (15) 1111 S, R AGN 1110 14 22
05 2 0010 UCN S (15) 1111 D, E GAN 1101 13 17
08 3 0011 CCN P (15) 1111 N, K AAN 1100 12 23
10 5 0101 GUN V (15) 1111 C, W UGN 1010 10 23 36 85
08 6 0110 ACN T (15) 1111 H, Q CAN 1001 9 22
04 7 0111 GCN A (15) 1111 Y, ct UAN 1000 8 15
17 11 1011 CGN R (15) 1111 I, M AUN 0100 4 24
01 15 1111 GGN G (15) 1111 F, L UUN 0000 0 27
30 G GG GG GGG GG GG GG G 88
(333 + 592 = 925) (1110 +1110) G 259 000
(36 + 88 = 124) (30 +85 = 115) xx
The designations: an M number of atoms within amino acid side chain; on1 M ordinal number in binary records; on2 M ordinal number in decimal records; c1 M codons with containing one- meaning nucleotide doublets; c2 M codons with containing two-meaning nucleotide doublets; aa M amino acids. One can notice here a determintion through a linear Fibonacci sequence: 1, 2, 3, 5 (LSPV with 36 = 4 x 9 atoms) and through a space Lucas sequence: 4, 7,11 (IMAR with 45 = 5 x 9 atoms). Within the non-Fibonacci spase and non-Lucas spase there are SRDENKCW = 85 and THQYFLG =73atoms, respectively. These two last series correspond with two other series with the same quantities of the number of atoms. [Cf. two series in Table 5.2: DENKCWHQ = 85 and SRIMFL = 73.]
9 I II III IV
V VI VII VIII
IX X
Figure 1. Correspondent molecules for 20 protein AAs (16 aliphatic and 4 aromatic) and/or for two and two (4 aromatic) Py-Pu bases. The designations: I. benzene; II. toluene; III. methane; IV. cyclopropane; V. pyridine; VI. pyrrole; VII. pyrimidine; VIII. imidazole, IX. indole, X. purine.
10 I II III IV
Figure 2. Atom groups and guanidine molecule, correspondent for some protein AAs. The designations: I. methyl group; II. isobutyl group; III. isopropyl group; IV. guanidine molecule.
2.2. The starting molecules and derivatives In Figure 1, we can see three "start" precursor molecules: benzene (I), methane (III) and cyclopropane (IV), the simplest hydrocarbons from the corresponding groups M arenes (the most stable hydrocarbons), alkanes and cycloalkanes, respectively. The potential candidates for builders are themselves, or their derivatives.4 By looking at the ten formulas in Figure 1, we find that aromatic hydrocarbons provide precursors for all four Py-Pu bases and all four aromatic AAs M the simplest aromatics, from the most stable six-membered and five-membered groups. On the other hand, observing the structure of the methane molecule (III), we note that here (in the act of selection), besides the principle of similarity, the principle of self-similarity applies: methane structure pattern is not only the pattern included into the "head" (amino acid functional group) of each of 20 AAs, than also in the body of all 16 AAs of the alanine stereochemical type. [Beside that the methan structural pattern (through the CH2 group, located between the head and the "body", i.e. side chain of AA) makes the basis of the alanine stereochemical type, the glycine type possess also the same pattern.]5
2.2.1. Derivative of derivative
Through the structure of alanine, the simplest amino acid of the alanine stereochemical type, all four aromatic AAs are included in a set of 16 AAs of alanine type. Phenylalanine, as its name suggests, is an alanine derivative by replacing a hydrogen atom in the side chain of alanine, in the CH3 group, with a benzene phenyl
4 The details about the precursors of AAs ca 3 )64' 4458 5 About four stereochemical types of AAs one can see in (Popov, 1989; Rako6evi4 < *)4' 4478
11 group. By this act appears the situation which is also readable as so that phenylalanine is formed as a derivative of derivative: in the benzene derivative toluene (II in Fig. 1), one hydrogen atom is replaced by an amino acid functional group. All together, the self- similarity of amino acid molecules in them-selves is realized.6 After the selection of phenylalanine (at the beginning of the first column of GCT) , as the first possible aromatic AA with a six-membered ring, its first possible derivative, tyrosine, is selected (at the beginning of the third column of GCT). It possesses the hydroxy group in the most stable para position (and not in a less stable ortho or meta position). Following two Mendeleev principles M the continuity and minimum of change M one should expect a derivative with a nitrogen functional group (amino group), instead of the oxygen (hydroxyl) group. But in such a case we would have a functional group with three instead with two atoms, what would not be in accordance with the principles of balancing and nuancing, which, in the case of genetic code, are also valid, as we will further show. If we see the beginning and the end of GCT as complementary to each other (see below), then we see that, as the first minimal change in the set of arenes occurs at the beginning, so does the very first minimal change in the set of alkanes. [It is, therefore, the complementarity of the two most stable classes of hydrocarbons; one that is very complex, with hybridized chemical bonds and delocalized electron orbitals (arenes) and others that are very simple, with simple chemical bonds and localized electron orbitals (alkanes).]7 As in the beginning, a minimal change occurs in which from Phe follows Tyr, so the same or similar process occurs at the end: the substitution of a hydrogen atom in Gly (last in the fourth column), with one methyl group (Comment 2), follows Ala (last in second column), which is "automatically" (as explained above) in relation to Phe. Moreover, the substitution of hydrogen atom in the Gly with an isopropyl group (III in Fig. 2) appears Val (last in the first column). At the "same time", together with valine was generated proline (Comment 3); and in a parallel process, in the first row, as a direct derivative of alanine, follow Ser, the first in second column, and Cys in the fourth one (Comment 4).
Comment 2. Within the side chain of Gly, as the first in the set of AAs, there is one atom only; in next amino acid, Ala, the 4 atoms. As we see, the numbers 2 and 3 are "forbidden" due to the four valences of the carbon. The question arises whether the set of number of atoms is chaotic or nonchaotic. Our researche shows that valid is the second response M not chaotic 3 )64' 2017a, Table 3, p. 16).
6 )64' >>9' ? : . "Hypothesis on a [prebiotically] complete genetic code (CGC): By this hypothesis ... we support the stand point that CGC must be based on several key principles. ... 1. The principle of systemic self-related and self-similar ; G@ [Note: This hypothesis best corresponds with the same such hypothesis of V.V. Sukhodolets (1985).] 7 "Contraria sunt complementa." (A motto at Niels BohrR own coat of arms, which featured a taijitu, symbol of yin and yang, designed in 1947. 12 Comment 3.The generating of proline is realized through isopropyl group, whose "triangle" is connected to the head of AA: in the case of valine with the vertex, and in the case of proline with the side. (For details see )64 < *)4' 4478 A0 0 ?B Comment 4. On a spatial-spheric model of the Genetic Code Table, the first and last rows are neighbors, as well as the first and last columns. This is, mutatis mutandis, as in LIGHT (Logical- Information-Geometric-Homeomorphic-Topological) model of the 3 )64' 1994, Fig. 4.1, p. 54).
Already with the selection of the first aromatic AA (Phe), we see the correspondence with two Py nucleotide bases, because the pyrimidine is a benzene derivative. As can be seen from Figure 1, the selection of a two-nitrogen pyrimidine (VII) is preferable than one-nitrogen pyridine (V). Two chemical reasons can be crucial here. Pyrimidine is more than a weaker base, but, more importantly, its far greater ability is to establish hydrogen bonds in potential dimers, which are actually found in natural DNA and RNA. One and the other is a distinct advantage for further balancing and nuancing. [As the "sowing" through the Darwin sieve, in the selection of chemical elements of the second period of the PSE, stopped before the fluorine, which, with its high reactivity, "burns" life before it arises, so this has also been shown here that the sieve was non-selective for stronger organic bases, which also lack the ability for balancing and nuancing in dimerization.]
2.2.2. Diversity enriching and economicity
With the selection of the third (Trp) and the fourth (His) aromatic AAs, the correspondence with two Pu nucleotide bases is also evident. In addition, it is also evident that in Darwin's prebiotic (chemical) evolution the selectivity of the sieve is "enriched" so that two additional principles apply: the principle of economicity and the principle of the enriching of diversity by increasing the degree of multi-meaning in relationships. The influence of both these principles is seen in the act of selection of tryptophan. In the case of Trp, we have a fusion of benzene with pyrrole into a two-ring indole (I + VI = IX), rather than with pyridine in double-strand quinoline (I + V = Quinoline); in the case of histidine, imidazole (VIII) is selected as similar to pyrrole (VI). In addition to the stated reasons for the non-ability of pyridines, the reason are also the two quoted principles: by selection a five-membered aromatic ring (in both cases, in the case of pyrrol, as well as imidazole), instead of the six-member, molecular diversity increases, and in addition, imidazole provide an aromatic electronic sextet which possess the benzene too; moreover, the selection of this new type of electronic sectet enriches the multi-meaning relations within the genetic code (Box 1). Both five-membered rings, pyrrole and imidazole, in fused compounds, within the constituents of genetic code, provide approximately equal acidity / basicity with a lower reactivity. Imidazole (VIII) is found in purine (X), in two purine bases, but also, as already mentioned, in the fourth aromatic AA histidine (the fourth in this discussion).
13 Additionally, one more point is needed here. When we discuss the analogy of six- membered and five-membered aromatic rings, apart from the similarities presented, we also mean the coherence of structural patterns (structural motives) and the similarity in the "flow" of delocalized electrons and electronic densities in the molecule. Thus, for pyrimidine (VII) and imidazole (VIII), except for the possession of an aromatic sextet, we say that they are analogs with the fact that they both have non-adjacent nitrogen atoms. [The pyrimidine is not analogous to the pyrazole (the imidazole isomer) because both the pyrazole nitrogen atoms are adjacent.]
Box 1. The economicity of aromatic five-membered rings L. G. Wade, Jr, Organic Chemistry, 8th International Edition, New York, 2013, p. 731: "Pyridine is an aromatic nitrogen analogue of benzene. It has a six-membered heterocyclic ring with six pi electrons. ... Pyridine shows all the characteristics of aromatic compounds. ... Because it has an avaiable pair of nonbonding electrons, pyridine is basic." Ibidem, p. 732: "Pyrrole is an aromatic five-membered heterocycle, with one hydrogen atom and two double bonds. ... Although it may seem that pyrrole has only four pi electrons, the nitrogen atom has a lone pair of electrons. The pyrrole nitrogen atom is sp2 hybridized, and its unhybridized p orbital overlaps with the p orbitals of the carbon atoms to form a continuous ring. The lone pair on nitrogen occupies the p orbital, and (unlike the lone pair of pyridine) these electrons take part in the pi bonding system. These two electrons, added to the four pi electrons of the two double bonds, complete an aromatic sextet." Ibidem, p. 733: "Imidazole is an aromatic five-membered heterocycle with two hydrogen atoms. The lone pair of one of one of the nitrogen atoms (the one not bonded to hydrogen) is in a sp2 orbital that is not involved in the aromatic system; this lone pair is basic. The other nitrogen uses its third sp2 orbital to bond to hydrogen, and its lone pair is part of the aromatic sextet. Like the pyrrole nitrogen atom, this imidazole NMH nitrogen is not very basic. ... Purine has an imidazole ring fused to a pyrimidine ring. Purine has three basic nitrogen atoms and one pyrrole-like nitrogen."
2.3. The branching structures Above we analyzed that part of the scenario that relates to the selection of Py-Pu bases and four aromatic AAs. Now we are going to analyze the generating, i.e. the selection of those AAs, which have a "pure" hydrocarbon side chain, a standard series (Ala, Leu, Val, Ile) and two AAs with a non-standard side chain (Gly, Pro)(Comment 5.) We assume that the logic of selection by the principle of matching the same or similar structural patterns here is also valid. Through analyzing these six AAs, we will continue to "keep on eye", for comparison, the remaining eight non-sulfur AAs (Section 2.3.1).
Comment 5. About four diversity / )64' > b, p. 822: (GP), (ALVI), (CMFYWH), (RKQNEDTS). Within the sub-set ALVI, all four AAs have a "pure" hydrocarbon side chain, and hence all four are nonpolar. Within the sub-set GP, for Gly we say that the non-standard hydrocarbon amino acid is hence because its hydrocarbon originates only from the head, and not from the body. (Hence its semi-polarity, as explained in Comment 7). The proline is also non-standard hydrocarbon AA, but through other reasons. By having three
14 methylene groups in the body, that is, in the side chain, it is a hydrocarbon AA; but because the sequence of three methylene groups binds to the head, it is not a "pure" hydrocarbon AA. From the same reason, such a structure makes proline a semi-polar AA. This insight was necessary for @ @ 3 )64' >>9' Tables 3 & 4 and Section 3.3).
2.3.1. The "mapping" from the head to the body
Observing the structure of methane, the simplest alkane (III, Fig. 1), we see that it is a form that we find in the amino acid functional group (in the "head" of AA), as already mentioned above. Even more than that, we note that the generating of the body (side chain) of the remaining eight AAs can be understood as a "mapping" of partial functional groups from the head to the body. And, the mappings are as follows: the amino group leads to the generating of Lys & Arg; hydroxyl to Ser & Thr; carboxyl to Asp & Glu; and, finally, the fusion of the carbonyl and amino groups forms two amides (Asn & Gln), the derivatives of the two carboxylic AAs.
2.3.2. The pure hydrocarbon AAs
We return to the problem of generating four standard hydrocarbon AAs (Ala, Leu, Val, Ile) and two non-standard (Gly, Pro). We continue to observe the structure of the methane molecule (III, in Fig. 1) and note that this structure does not resemble the structure of the following members of the homologous series of alkanes (ethan and propane). Paradoxically, the methan looks like the fourth member, not as n-butane but as iso-butane. Despite the fact that, after Ala, as methyl derivatives, it is expected that AAs, which are ethyl and propyl derivative, are selected, this, however, does not happen; such AAs exist not in the set of protein AAs. In Figure 2, the structures of the methyl and iso-butyl groups (I, II, in Fig. 2) are given. We see that the accordance is complete; the only difference is the size of electronic density on three "branches". Therefore, it becomes understandable why, after alanine, with the simplest hydrocarbon sequence of one methyl group, the expected derivatives with the atomic groups of ethyl, n-propyl, n-butyl come not, but iso-propyl and isobutyl derivatives, as well as one nitrogen analogue (II, III and IV in Fig. 2). [The nitrogen analogue is in arginine, by analogy of the iso-propyl group and the guanidine molecule.] Iso-butane derivatives are the only two isomeric AAs, Leucine and Isoleucine; Leucine as the second AA in the series of AAs of the alanine stereochemical type and Isoleucine as a second one in series of AAs of the valin stereochemical type. [The generating of the first AA in the alanine type (Ala) on the methane pattern was explained above; also of the generating of the first, and only one AA in the glycine type (Gly). The explanation, however, for the first AA in the valine type (Val) and the first, and only one in proline type (Pro) on the iso-propil pattern, we gave in one of the previous papers (see Comment 3).] 15 3. NUCLEOTIDE TABLES AND THEIR FACES 3.1. The 24 permutations of four Py-Pu bases In the set of four Py-Pu bases (UCAG) there are 24 permutations, each with three possible sequencing: 16 of the same bases in the first, or second, or third codon position. Of the 24 permutations, the chemical hierarchy (from the aspect of the complexity of molecules) corresponds only to the first (UCAG), which has a very distinct Py-Pu hierarchy, and all others, if it makes sense to analyze, then only in relation to the first permutation. Such permutations are UACG, with a strict hierarchy: two vs three hydrogen bonds; and CAUG, also with a strict hierarchy: amino vs. oxo functional group. The half-inversion UCAG/CUGA has no chemical meaning because "C" is a more complex molecule than "U". The same goes for UACG/AUGC, because (for an equal number of hydrogen bonds, A = 2 and U = 2), "A" is a more complex molecule. In the third case, the half-inversion CAUG / ACGU has a chemical sense. In the first case at the beginning is Py molecule "C" as less complex; in the second one, the Pu molecule "A" as poorer in the functional groups than Py molecule "C". [Adenine does not have two but only one functional group (amino group).] Chronologically, "an alternative model of translation" within genetic code was presented via ACGU permutation 3" &4' 445C " &4 < )64' >>6, 2006, 2007); and then a "p-adic model of ... genetic code" via CAUG permutation (Dragovich et al ., 2006, 2010, 2017). Bearing in mind, once more, that there are 24 permutations of four Py-Pu bases, each with three forms, from that aspect, F. Crick (1966, 1968) must first be acknowledged, which in the act of revealing the complete GCT, presented the best possible solution, from the aspect of the physico-chemical properties of the molecules, the builders of the genetic code (Crick, 1966, 1968). But the acknowledgement to R. Swanson that she realized that Py-Pu distinctions in the Gray Code model of the genetic code, and consequently on the binary tree, as well as in GCT, also (from the physical-chemical aspect) must follow Crick's model (Swanson, 1984).8 [Today we can say that a large number of works have been confirmed the Crick's model, from which we list here only some, which, mutatis mutandis, correspond with this our work (Brimacombe, et al., 1965; Nirenberg, et al., 1966; Woese, et al., 1966; Rumer, 1966; Swanson, 1984; Doolittle, 1985; Sukhodolets,1985; Leunissen and De Jong,1986; Brains, 1987; Taylor & Coates, 1989; Alvager et al., 1989; Koruga,1992; Shcherbak, 1994, 2008; Négadi, 2009, 2011, 2014; Castro-Chavez, 2010; Petoukhov, 2016).] 3.2. The nucleon number balances Starting from the said Crick's GCT model and the Swanson's Py-Pu distinctions, it was easy for us to show that these rules follow not only for the distinctions for 64 codons
8 About chemical sense of 24 permutation of UCAG see also in ( )64' >>:' 8
16 on a six-bit binary tree 3 )64' 445a), but also for the binary records of 16 nucleotide doublets (Table 1.2). On the other hand, it was possible to show that only in the GCT constructed via the first permutation UCAG, with four columns (NUN, NCN, NAN, NGN), each column with 16 codons, it can be shown in the GCT two significant nuance balancing splittings.
In the first 3 )64' >>9, Fig. 5, p. 226), the GCT is diagonally (and symmetrically) splitted into the upper part with 12 amino acid molecules (in which there are 888 nucleons) and the lower part with 11 molecules, which in the side chains possess 555 nucleons (here: Figure 3, on the left). In the second, nuance-balancing splitting 3 )64' >>7, Fig. 7) (here: Figure 4)], the GCT splits into 9 four-codon spaces, where 9 single amino acids are encoded, and 7 four-codon spaces where are encoded 7 pairs of AAs. For better and more light the current analysis, we give here a complete legend for the first splitting, which leads to a sophisticated nuancing and balancing:
"The nucleon number balance within two complementary parts (sub-systems?) of Standard Genetic Code Table. Up, in 12 AAs molecules (side chains), there are 888 nucleons (in first nuclide): F91 + L57 + S31 + Y107 + C47 + W130 + P41 + H81 + Q72 + R100 + S31 + R100 = 888. Down, in 11 AAs molecules (side chains), there are 555 nucleons: G01 + E73 + D59 + K72 + N58 + A15 + T45 + V43 + M75 + I57 + L57 = 555. [Cf. the possible relations with patterns (quantums) in the system of four-codon AAs (888M555=333) and non-four-codon AAs (555+555=1110)' 0 )' 449' D ' 9:6B@ 3 )64' >>9' . to Figure 5, p. 226).
This interpretation, given 14 years ago, today we are able to expand and supplement. We read as follows. If one diagonal splits GCT into the upper part with 12 amino acid molecules (FLSYCWPHQRSR) and the lower part with 11 molecules (LIMVTANKDEG) [here: Figure 3, on the left], then in 12 molecules, in their side chains, has 888 nucleons, and 11 molecules have 555 nucleons. The sense of the balancing, through harmonization with two Mendeleev principles (continuity and minimum change) is this: 555M666M777M888, where two external patterns "play", and two internal ones do not.9 And the same, "just a little different" (Mendeleev's continuity and minimum change), we find in the orthogonal splitting of GCT into four columns and four rows: within 1 & 4 rows as well as 2 & 3 columns of GCT there are 654 nucleons, while in 2 & 3 rows as well as in 1 & 4 columns 789 nucleons. In the question is, then, the sequence 456M789 / 987M654, in which in one case "plays" the original, and in the second one the mirror image (Verkhovod, 1994; )64' >>7' D :' 98 3 = Figure 4).
9 More than a curiosity: the sum 555 + 888 = 1443 (number of nucleons within side chains of 23 AAs) corresponds with the sum of the first four perfect numbers: 6 + 28 + 496 + 8128 = 6 x 1443. [About the hypothesis that the perfec 3 )64' 44:' 7>8 3 )64' > :' E :' F9?8.]
17
12 molecules . 12+1 molecules (119+20) at & 888 nu . (119+10) at (888-49) nu UU F UC S UA Y UG C . UU F UC S UA Y UG C L W . L W . . CU L CC P CA H CG R CU L CC P CA H CG R Q Q
AU I AC T AA N AG S AU I AC T AA N AG S M K R M K R
GU V GC A GA D GG G GU V GC A GA D GG G E E 11 molecules 11-1 molecules (120-20) at & 555 nu (120-10) at & (555+49) nu
Figure 3. Left side: The illustration is the same as in ( )64, 2004, Figure 5, p. 226), except one difference: in that previous paper in the question is only the number of nucleons, but in this chapter both M the number of atoms (at) and the number of nucleons (nu). Right side: It is all the same as on the left side, except here are relations via right diagonal.
18 7+7 AAs MOLECULES [2 x (80-1)] atoms UU F UC S UA Y UG C L W nucleons nucleons 654 CU L CC P CA H CG R 789 Q
AU I AC T AA N AG S M K R
GU V GC A GA D GG G E 9 AAs MOLECULES
[(128 x 1) 01] STOP codon (bases) atoms [(128 x 2) 10] AAs codon (bases) atoms [1 x (80+1)] atoms AAs 1-4 rows, 2-3 columns: AAs 2-3 rows,1-4 columns:
3456
Codon (nucleotides) atoms 3456
Figure 4. The number of atoms and nucleons within GCT. The number of atoms and nucleons in the classes and subclasses of amino acids corresponds to the parts of sequences of natural numbers, with principle of minimum (unit) change. The external vertical row on the left: the number of atoms in three "stop" codons, calculated to bases (U = 12; C = 13; A = 15; G = 16). The internal vertical row on the left: number of atoms in the remaining 61 codons, which have amino acid meaning. Two vertical rows on the right: the number of nucleons in the GCT according to (Verkhovod, 1994, Figure 2). Dark tones: two-meaning nucleotide doublets, each with two encoded amino acids [in total: two times (8-1) amino acids with 123+35 atoms]. Light tones: one-meaning nucleotide doublets, each with one encoded amino acid [in total: (8+1) molecules, with 81 atoms]. The bottom of Figure shows that in two and two columns, as well as in two and two rows, there are 3456 atoms in the codons, calculated to nucleotides (UMP = 34; CMP = 35; AMP = 37; GMP = 38). Notice that the distinctions into 9 single AAs (white areas) and 14 doublet AAs (dark tones) is the same as in Table 5.2.
By comparison Shcherak's findings in Rumer's Table of nucleotide doublets (Shcherbak, 1994, Figure 1) with our findings in the standard GCT (Rako64' >>9' Figure 5, p. 226), we see the quantitative relationships as follows. [By comparison, the left and right side in the Rumer's Table it will be observed, as well as the upper and lower diagonally distinct parts in GCT (here: Table 2.1 vs Figure 3, left side).] On the right side of the Rumer's Table, in the "16-1" non-four-codon AAs, in their side chains, as well as
19 in their heads, there are 1110 nucleons each, while in the diagonally distinct lower part of the GCT it has exactly one half of that quantity (1110: 2 = 555); if this quantity is added to the total number of nucleons in 8 ± 0 four-codon AAs (existing on the left side of the Rumer's Table), in their side chains (555 + 333 = 888), the quantity 888 is obtained, as it is the exact number of nucleons in the upper diagonally separated part of the standard GCT. (In both halves there are 555+888 = 1443 nucleons.) Bearing in mind that the above quoted result (from our previous paper) for the number of nucleons is obtained by separating via left diagonal, in this chapter we go a step further, analyzing the state of the number of nucleons in relation to the separation via the right diagonal too. In addition, we analyze the state of the number of atoms in both types of separation (Figure 3). For the number of nucleons, we obtain the result 555 + 49 and 888-49, which is also a kind of uniqueness, as demonstrated in Survey 1. [Results for the number of atoms in both separations are presented directly in Figure 3.] For further evidence of the existence of correspondence in nuancing and balancing between the nucleotide doublet and the nucleotide triplet Table, it is also important to note the other uniqueness of the sequence 555-333-888, which had not been previously discussed (Survey 1); and also to notice that our series 555-333-888 (in form 333-555- 888) is included in the set of multiples of Shcherbak's "Prime quantum 037" (Survey 2).Survey 1. The uniqueness of the sequence of the number of nucleons in the two diagonally separated halves of GCT in Figure 3
(1221) (999) 222 206 216 16 = (4)2 (999) (555) 444 417 427 27 = (5.20)2 (777) (111) 666 628 638 38 = (6.16)2 (555) (333) 888 839 849 49 = (7)2
3,5,8 5,8,3 19 29 39 49 = (7)2 3,8,5 8,3,5 5,3,8 8,5,3 59 69 79 89 99
(0,1,1), (1,2,3), (3,5,8), (8,13,21), ...
The uniqueness of the sequence of the number of nucleons in the two diagonally separated halves of GCT [555 + 888 = 1443 in relation to the sequence (555 + 49) + (888-49) = 1443] manifests itself through similarity and self-similarity (849 vs 839) as well as through uniqueness of the number 49. Also, through the relation with the Fibonacci series (down, left), observed in the form of triples with one overlap. This insight is significant if we already have evidence that the GC was determined by the Golden mean, contained in the Fibonacci series ( )64, 1998a: Binary tree on Figure 1, contained "golden" AAs in relation to Farey tree on Figure 2, contained Fibonacci series as the key determinant). 20 Survey 2. The multiples of "Prime Quantum 037" and to it correspondent numbers
27 78 9 858 99 8991 999 26 78 26/3 858 286/3 8658 962 25 75 25/3 825 275/3 8325 925 24 72 8 792 88 7992 888 ... 16 48 16/3 528 176/3 5328 592 15 45 5 495 55 4995 555 ... 10 30 10/3 330 110/3 3330 370 9 27 3 297 33 2997 333 8 24 8/3 264 88/3 2664 296 7 21 7/3 231 77/3 2331 259 6 18 2 198 22 1998 222 5 15 5/3 165 55/3 1665 185 4 12 4/3 132 44/3 1332 148 3 9 01 66 11 999 111 2 6 2/3 66 22/3 666 074 1 3 1/3 33 11/3 333 037 1/3 11/3 111/3 @0 @ 1st 2nd 3rd
?' ??' ??? F?' F??' F?' 0 )R "Prime Quantum 037". The explanations in the text.
21 Survey 3. "Golden" amino acids, their complements and non-complements
F 14 15 Y L 13 (66M1) 04 A Q 11 08 N
P 08 13 I T 08 11 M (60+1) S 05 05 C G 01 10 V
D 07 10 E K 15 (78 ± 0) 17 R H 11 18 W
Presented is the modified Survey 2.1, given in )64' 445' 54 G/' @$ @ amino acids (dark tones) are given in the order they have on the binary tree; and here according to the growing mass of their molecules. Over the "golden" AAs, there are their complements, and at the bottom non-complements. The number of atoms in side chains of amino acids is as follows: GSTPQLF = 60, VCMINAY = 60 + (1 x 6) = 66, DEKRHW = [60 + (1 x 6)] + (2 x 6) = 78. Here, however, is shown that one other distinction is possible with the same patterns of the number of atoms (self-identity!); such a change that leads to a minimum change, for ± 1.
With the insight in Survey 2, we find the following relationships. The quantities of the number of nucleons that are valid for the lower and upper parts of GCT, 555 & 888, are in the immediate neighborhood of the quantities 592 & 925, respectively, valid for four-codon AAs. In other words, the difference is exactly for one "Prime quantum 037" (592 -555 = 037 and 925 - 888 = 037); with a valid neighborhood also for two referent points: (592 - 333 = 259 and 555-333 = 222) (259 - 222 = 037). As we see, nucleotide doublet and triplet Tables are balanced not only within themselves, but they are balanced one to other, through the Pythagorean triple: (3 ^ 2) x 037 = 333; (4 ^ 2) x 037 = 592; (5 ^ 2) x 037 = 925; (592 - 333 = 259).10
10 Regardless of Shcherbak's work, which announced that the multiples of OH I >?:@ (in form of a Table of number triplets) appear to be a determinant of the Genetic Code, I found the same Table but with the number singlets (analog to Table in Survey 2). I presented the Table at Scientific conference of the Montenegrin Academy of Sciences and Arts in Cetinje, 27-30. September 1993. An integral article was published in Proceedings of Scientific Conferences of the Montenegrin Academy of Sciences and Arts 22 3.3. The atom number balances For the analysis of the relationships of the nucleotide doublets and the nucleotide triplets Table, we start with the Rumer's double Table (Table 2.1) and we compare it first with Tables 3 & 4, in which the codons from Table 1.1 (in the form of nucleotide doublets) are mapped. We see that the separation into the upper and lower half of the Table 2.1 is dictated by hydrogen bonds, while in Table 4 it is dictated by the positions of the doublets on the binary tree 3 )64' 445' D 8. This is important because in Table 5.1, which is mapped from Table 4, we have the same balance of the number of atoms in the corresponding amino acids as in Table 2.1 (120: 119); all that in situation when the distinctions between the amino acid molecules are dictated by the polarity.
Table 5.1. The horizontal CIS display into one-meaning and two-meaning nucleotide doublets and corresponding amino acids (I)
1 23 567 11 15 L S P V T A R G +3.8 -0.8 -1.6 +4.2 -0.7 +1.8 -4.5 -0.4
-0.8 -4.5 -3.5 -3.5 -3.5 -3.9 +2.5 -0.9 -3.2 -3.5 -1.3 +4.5 +1.9 +2.8 +3.8 S R DE NK C W H Q Y I M F L 14 13 12 10 9 8 4 0
Final LARG = 36-1 35+85 = 120 SPVT = 30+1 result SRYIMFL = 88 88+31 = 119 DENKCWHQ = 85
All is the same as in Table 4, with an additional distinction of amino acid molecules through their polarities, measured by the hydropathy index (Kyte & Doolittle, 1982). Notice a diagonal balance in which it is shown that number of atoms in amino acid singlets and doublets is determined by the middle pair of the total number of atoms within the set of 23 AAs (0, 1, 2 , 3, 4, ..., 119-120, ..., 235, 236, 237, 238, 239); all that in a strict distinction into polar and non-polar AAs.
(CANU), 1995, Book 35, Department of Art CANU, Book 12, pp. 245-265 (as Table 2). The Shchrbak's Table I found just in the year in which it was published (Shcherbak, 1994). 23 Table 5.2. The horizontal CIS display into one-meaning and two-meaning nucleotide doublets and corresponding amino acids (II)
1 23567 8 11 15 L S P V T A Y R G +3.8 -0.8 -1.6 +4.2 -0.7 +1.8 -1.3 -4.5 -0.4
-0.8 -4.5 -3.5 -3.5 -3.5 -3.9 +2.5 -0.9 -3.2 -3.5 ±0.0 +4.5 +1.9 +2.8 +3.8 S R DE NK C W H Q ʿ I M F L 14 13 12 10 9 8 4 0
F.R. LAYRG = 50 50 + 31 = 81 SPVT = 31 II SRIMFL = 73 73+85 = 123 + 35 DENKCWHQ = 85
All is the same as in Table 5.1, with an exception: the amino acid tyrosine is above instead down. Notice here a specific balance in which it is shown that the number of atoms in amino acid singlets (9 AAs above) equals 81 as in 10 AAs of class II, handled by class II of enzymes of aminoacyl-tRNA synthetases; On the other hand, the quantity of the number of atoms in 14 doublet AAs is 123 + 35, where the number 123 corresponds to the number of atoms in class I of AAs, handled by class I of enzymes aminoacyl-tRNA synthetases and the number 35 is a "surplus" of 35 atoms is a "balance fraction" which, when passing from 204 atoms in the set of 20 AAs to 239 atoms in the set of 23 AAs, corresponds to the quantity for three doubled AAs (L13 + S05 + R17 = 35). About classification into two classes of AAs, handled by two classes of enzymes aminoacyl-tRNA synthetases, one can see in: Wetzel, 1995; Rako64' 44:C Rako64' 445' 0/ 9' 4> [Cf. two significant series DENKCWHQ = 85 and SRIMFL = 73 with two self-similar series (SRDENKCW = 85 and THQYFLG =73 atoms, respectively, in Table 4.]
It is obvious that in the quantities "120 atoms" and "119 atoms," in two Tables, not all AAs are the same, so it is important to analyze what the difference is. Within the quantity "120 atoms" in Table 2.1 there are T+V+S+S = 28 and within Table 5.1 it is A+G+N+K = 28 atoms. Within both Tables there are L+R+D+E+C+W+H+Q = 92 atoms; altogether 28+92=120 atoms. On the other side, within the quantity "119 atoms" we have a vice versa situation: in Table 2.1 there are A+G+N+K = 28 atoms and in Table 5.1, it is T+V+S+S = 28 atoms. Within both Tables there are P+R+F+I+Y+L+M=91 atoms; altogether 28 + 91 = 119 atoms. The difference between "quantity 92" and "quantity 91" comes from two different sequences, which differ by one molecule and one atom: D+E+C+W+H+Q=62 and P+F+I+Y+M = 61 atoms. The same sequence in "quantity 92" and "quantity 91" is the sequence L+R = 30 atoms. If we bring this fact in relation to the upper sequence T+V+S+S = 28, it becomes clear that the amino acids L, R and S are duplicated in GCT just for balancing and nuancing.
24 Following these comparisons, it makes sense to compare the "faces" of the nucleotide doublet Tables (Table 2.1 and Table 5.1) with the "faces" of the nucleotide triplet Table (Table 1.1, Table 1.3 and Figure 3). We see that the nuance-balancing, as in each individual Table, and among them, is determined by the middle pair of the number of atoms in the side chains of the AAs, in a set of "23" AAs, by shifting "the tongue on the scale" by ± 10 in the first as well as for ± 10 in the second step. [We can split the number of 239 atoms (within the side chains of 23 AAs) into the pairs, where 119-120 is the middle pair (0, 1, 2, 3, 4, ..., 119-120, ..., 235, 236, 237, 238, 239).]11 Bearing in mind the fact that the number of atoms determines the number of nucleons, it is necessary, with the given results, to give one note more. In both cases, Shcherbak's as well as Verkhovod's result on the scene is a "block" exchange of quantities, analogously to that we first presented in the work on essential and non- essential AAs ( )64 & *)4' 447), and later in other works. Future researches should show if these "block" exchanges remain within the limits of nuancing and balancing, or, additionally, they themselves "represent analogies with quantum physics" (Shcherbak, 1994), as we stated in the Introduction.
3.4. The sense of the modification R In a #) 3 )64' > ?8' # # @ @ nucleotide doublets from the Modified Rumer's Table (Table 2.2) into GCT corresponds to the nuancing balance of polarity of the correspondent AAs, measured by the Cloister energy parameter (Swanson, 1984).12 Thus, each first doublet, of four quartets, within the Modified Rumer's Table occupies one of the four (4±0) external four-codon-spaces in GCT, filled with nonpolar AAs: GG M Gly, UU M Phe & Leu, GU M Val, UG M Cys & Trp; each second doublet occupies one of the four (4±0) internal four-codon-spaces, filled with polar AAs: CC M Pro, AA M Asn & Lys, AC M Thr, CA M His & Gln; each third plus forth doublet occupies one of the four plus four [(4+1) & (4-1)] intermedial four-codon-spaces, filled with (4+1) polar, and (4-1) non-polar AAs; polar: UC M Ser, UA M Tyr, CG M Arg, AG M Ser & Arg, GA M Asp & Glu; non- polar: GC M Ala, AU M Met & Ile, CU M Leu. Intermedial amino acids are listed in a continuous circular order, starting from Serine with the lowest value for polarity (the lowest value in the set of 5 intermedial and polar AAs), and going in the clockwise direction. The nuancing-balancing as follows: S= 0.24; Y = 0.42; [R=0.87; S=0.24; R=0.87] (R+S+ R) /3 = 0.66; D = 0.69; E = 0.71; A = - 0.09; M = - 0.57; I = - 0.56; L = - 0.54 (Notice that A & L are the first and the last, respectively) (Comment 6).
11 This again resembles "The Little Gauss" algorithm in the process of adding numbers from 1 to 100, which was given to a nine-year-old Gauss by his teacher ( )64, 2011b, p. 833). 12 Cloister energ/ O / G J / / /F / because it is an in situ measure of the property of interestP (Swanson, 1984).
25 External amino acids can also be listed with a growing sequence of values. Namely, the nuancing-balancing is as follows: C= - 0.73; F= - 0.56; V= - 0.52; G= - 0.16; W= - 0.25. As we see, the exception is where the listing has been started: Trp is more non-polar than Gly (Comment 7). Internal amino acids cannot be listed in a rising sequence, but via the "broken" line, the zigzag periodicity: H = 0.00; T = 0.27; P = 0.46; N = 0.52; Q = 0.91; K = 1.46. As we see, here there are His as first and Lys as last in their column (the first and last in the set of internal AAs); Thr as the next is on the left in the previous column; Gln as next in the column; Gln vs Pro as the next on the left; and, finally, Asn as the next in its column (Comment 8). Comment 6. In the set of four intermedial non-polar AAs, the exception in the falling series of values of the cloister energy parameter is Ala. It is signficant that Ala is also an exception by its specificity of molecular structure within the set of 16 AAs of the alanine stereochemical type.
All of them are characterized by the existence of the CH2 group between the head and the body. "Reading" so for alanine, it turns out that it has a body of glycine. On the other hand, if the hydrogen atom in the glycine is substituted by isopropyl group, a valine (valine type) is formed. However, as Valine is an Oinver P of proline, as a result we have that all four types are interconnected, what represents their similarity and self-similarity. Comment 7. This exception at the end of the non-polarity sequence, expressed in relation to Gly-Trp, corresponds with the exception of the polarity of these two AAs when the polarity is measured by the index of hydropathy. Namely, these two AAs by the hydropathy are not non- polar but polar (cf. their hydropathy values in Table 5.1). The reason for this disagreement lies in the fact that (by their chemical structure) both AAs are semi-polar. Namely, Trp has at the same time a nonpolar benzene ring and a polar pyrrole ring. On the other hand, the side chain of Gly is a hydrogen atom that is nonpolar. But this atom, together with another one, is at the same time in the head of AA. Hence the polarity. All together, it results in semi-polarity. Comment 8. We said that the internal amino acids are polar. However, it would be more correct to say that they are not nonpolar. It is because of His who has a zero value. It is, therefore, an exception; and it is an exception in the whole set of canonical AAs, since the only one contains only a five-membered aromatic ring. [This is not the case with Trp, because it has a six- membered benzene ring together with the five-membered pyrrole one.] The nuancing-balancing through polarity, presented above, are followed by the nuancing-balancing of the number of nucleons and atoms. Irrespectively of the modified Table of nucleotide doublets, V. shCherbak (2008, Fig. 10, p. 173) showed that it makes sense a codons display into GCT exactly as we shown here for nucleotides doublets: four squares at the corners and four squares in the center; then, eight squares in middle, i.e. in O#P shE )R #' the next: AAs in four squares in the corners as well as AAs in four squares in the center have 369 nucleons each [(F91 + L57 + V43 + G01 + W130 + C47 = 369); (P41 + T45 + K72 + N58 + Q72 + H81 = 369)].
26 X shE )R ' # # = quantity give the AAs in right site, in relation to the diagonal FMG, in the spaces of the doublets UC, UA, CG and AG (S31+Y107+R100+S31+R100 = 369). On the left side of the diagonal (in spaces CU, AU, GC and GA) there are 336 nucleons (L57+I57+M75+A15+D59+E73=336), what means 33 nucleons less, in relation to 369. With this emergence of difference of "33" on the sce ne appears a specific self-similarity because the number 33 is an important determinant of the number of atoms in the rows and columns as we shown in Table 2.1, down (Comment 9). Comment 9. If we consider the set of "61" of AAs, then in the rows, YNR & RNY, there are 8 x 33 = 264, and in the others, YNY & RNR, 10 x 33 = 330 atoms. On the other hand, in two pyrimidine columns, NYN, there are (9 x 33) M 1 = 297-1, and in two purine ones, NRN, (9 x 33) + 1 = 297 of atoms (Rak64' >>9' ?' 98 3K / ' + all four types of nucleotides). [Notice that the numbers 264, 297 and 330, we can find also in a unique Table, presented in Survey 2 (positions: 8, 9 and 10).] To these nuancing-balancing by nucleon number we now also add the nuancing- balancing by the atom number: AAs in four squares in the corners as well as in the center of GCT have 61 atoms in amino acid side chains (F14 + L13 + V10 + G01 +W18 + C05 = 61); (P08 + T08 + K15 + N08 + Q11 + H11 = 61)]. In relation to the diagonal FMG there are 58 and 59 atoms, respectively; on the left: L13 + I13 + M11 + A04 + D07 + E10 = 58, and on the right: S05 + Y15 + R17 + S05 + R17 = 59. These quantities (58 and 59) are the same as the quanti / 0) R / 3# further nuancing-balancing): 58 in two inner and 59 hydrogen atoms in two outer rows 30) ' 456C )64' > :' 5?>8 [Notice that 58 + 59 = 117 is total number of hydrogen atoms in 20 canonical AAs of GC, within their side chains, what is the nuancing-balancing once more.]
4. A SPECIFIC CHEMICAL COMPLEMENTARITY AS THE CIPHER
Bearing in mind the Gray Code model of the genetic code, Binary-code tree and the Scenario, presented in Chapter 2, just their possible connection, a generalized standard Genetic Code Table (as Table 1.1) can be generating; also that a sophisticated analysis of the Rumer's nucleotide doublet Table is possible (as in form of Table 2.1), as well as a modification of Rumer's Table (as in form Table 2.2); such a modification that the modified form with the original makes the inseparable whole. On the other hand, it was proved by a justified and necessary of generating another arrangement that would represent a unifying arrangement of the binary code tree and GCT (Table 3 in relation to Table 4). Such an arrangement makes sense to be named CIS (Canonical Invariant System) since the canonical amino acids in it are strictly determined and possess positions which it cannot be changed.
27 In the next step it makes sense to bring EJ0 R able, so that the one-meaning doublets will be separated from the two-meaning ones (Table 4); such a procedure in order to analyzing the interrelations of splitted doublets from the aspect of two types of arrangements: according to their positions in the CIS and according the number of their hydrogen bonds.
Now about the interrelation between Table 2.1 and Table 2.2. Both tables contain four doublet quartets. In both Tables, the two upper quartets start with GG/UU doublets, and the two lower ones start differently: with AC/CA at first and with GU/UG in the second case; we say that in the first case the lower quartets follow by dissimilarity, and in the second case by the similarity of nucleotide bases. In addition, the order of the remaining two and two doublets, as well as the corresponding amino acids, have also changed; in total the changes are subject to three pairs. Thus, in Table 2.1, we have ThrMVal, ArgMAla and SerMLeu, whereas in Table 2.2 there are: ValMThr, AlaMArg and LeuMSer. [Observed by individual amino acids, their order in Table 2.1 better suits to the order in the binary tree (reading in reverse order); however, observed by the amino acids pairs, better suits the order in Table 2.2. (Cf. order of AAs in Table 5.1).]
4.1. Three types of distinct chemical complementarity Observing the Table of nucleotide triplets (Table 1.1) and the Table of nucleotide doublets (Tables 2.1 & 2.2), we note that the pairing and/or interconnection of the nucleotides is characterized by three distinct chemical complementarities: 1. Being Py or Pu, 2. The Py-Pu interconnection and/or pairing with two, or with three hydrogen bonds, 3. The pairing over oxo or amino functional groups. All three distinct complementarities are expressed into three key systems (the three modes of presentation) of the Genetic code: 1. In the standard GCT (Crick, 1968), 2. At Gray code model of GC (Swanson, 19898' ? G (/ $E 3 )64' 455' D ? ' >C 445' Figure 1, p. 284). The third type of complementarity (Py-oxo with Pu-oxo; and Py-amino with Pu- amino) and the third type of presentation of genetic code (on the Binary-code tree) appear now to be the key to seeing the existence of the genetic code cipher and its key. Namely, the cipher key represents the positions of the nucleotide doublets and their associated AAs on the binary tree, and the cipher itself represents the specific complementarity of both M the doublets and the amino acids (Comment 10). In addition, complementarity follows in this order: 0-15, 1-14, 2-13, ..., 6-9, 7-8, as shown in Tables 1, 3 and 4. The complementarity described herein ("external complementarity" ) on the hypercube (B4),
28 as a Boolean model,13 can be understood as summing up until the same sum, to the number 15, or up to the number 1111 in the binary record (middle part in Table 4).
Comment 10. In this hierarchy the term (and notion) "genetic code" remains what has been from the beginning: a connection between four-letter alphabet (four Py-Pu nucleotides, in form of codons) and a twenty-letter alphabet (twenty amino acids).
4.2. Nuancing and balancing of chemical structures and properties The specificity of the chemical complementarity, which we are talking about here, is expressed as complementarity of the nucleotide doublets as well as the corresponding amino acids on each two vertices on the Boolean hypercube (B4) whose binary values give the sum 1111 in the binary record (corresponding to the number 15 in decimal one). In the question are the nuanced and balanced chemical complementarities, both through chemical structures, and through the chemical properties of molecules. Examples are already listed in the second section of this chapter (in the Scenario), and here we again present some characteristic examples in the new meaning (cipher M key of the cipher). Thus, we see that the initial (zeroth) doublet UU is complementary to the last doublet GG, which is the complementarity through the chemical difference.
4.2.1. The chemical hierarchy in the set of four nucleotides
Of the two pyrimidines, uracil is simpler because it has two the same functional groups (oxo functional groups), while cytosine possesses two different functional groups M oxo and amino. Of two purines, glutamine is more complex because it possesses both of these groups, while adenine has only one amino group.14 Paradoxically, the simplest doublet, UU, encodes Phe, very complex AA, in whose side chain is very complex benzene ring. On the other hand, the most complex doublet, GG, encodes Gly, the simplest within the set of 20 protein AAs, with only one hydrogen atom in side chain.
4.2.2. Similarity in Dissimilarity
We might say that complementarity, shown above, is complementarity through dissimilarity. However, a deeper chemical insight shows that this complementarity is a complementarity through similarity in dissimilarity. In the case of two dissimilar nucleotides (as Py vs Pu), UU vs GG, the similarity is that both possess the oxo group; in the case of two dissimilar AAs (Phe vs Gly), the similarity is in a similar structural pattern within both molecules. The matter becomes clearer when it is seen that the doublet UU does not encode only Phe than Leu (formulas II and III in Figure 1, in
13 The details on how GC can be understood as Boolean space can be seen in two of our studies ( )64, 1994, 1997b), both installed on our site. 14 These chemical distinctions are important from the aspect of the answer to the question of which form of GCT is the best (cf. Section 4.3.1).
29 relation to formulas I and II in Figure 2). [In reality the codons code for amino acids. Thus, Gly is encoded with the four most complex codons (GGN). On the other hand, chemically very complex AA, Phe, is encoded with only two simplest codons, UUU & UUC.]
Within the CH3 group of toluene, one hydrogen atom is substituted by an amino acid functional group. Hence, a H-C-H group is formed between the head and the body. In this case, a C atom from the amino acid functional group and a C atom from the benzene ring are bonded vertically for the C atom in H-C-H group. All together, a form analogous to structures I and II in Figure 2 is obtained. As we see, in the case of nucleotides, the dissimilarity is derived from the difference between their two dissimilar types (pyrimidine vs purine), and the similarity is derived from the same functional group. In the case of AAs, the dissimilarity arises from the difference of molecular classes involved in the construction of the amino acid side chain, and the similarity commes from the similarity in molecular structural patterns.
4.3. The chemical complementarity as a neighborhood logic In order to be able to analyze other examples of complementarity in the standard GCT, we must first answer the question of which GCT arrangement must be presented; in other words, which of 72 possible the arrangements, is the best; the best in terms of respecting the chemical hierarchy, from the aspect of less or more molecular complexity. It is only after answering this question that it makes sense to look at the state of chemical complementarity of nucleotide and/or amino acid molecules (Sections 3.1 and 4.3.1). The correct answer to the raised question was already given in 1966 in the form of the above presented tables, the Nucleotide Triplet Table (Crick, 1966, 1968) and the Nucleotide Doublets Table (Rumer, 1966). In the presentation of these Tables, both authors went from experimental facts, which in practice, mutatis mutandis, already led to the Genetic Code Table (Brimacombe et al, 1965; Nirenberg et al., 1966), in which "the best present version of the code is shown", as Crick wrote in his second paper on GCT (Crick, 1968, Section:" The Structure of the Present Genetic Code ", p. 367).
4.3.1. The best possible permutation arrangement
The meaning of the above-quoted E)R sentence is that only with the arrangement of the first permutation (UCAG) and codons with the same base in the middle position (16 times), the logic of the neighborhood within the four columns is achieved: 1st NUN, 2nd NCN, 3rd NAN, 4th NGN. Only in this case neighboring codons are mutually similar, and they can encode the same or similar AAs. And what was in force in 1966 and 1968, is valid today, with all in the presence the so-called deviant codes (Box 2). This is truly the best arrangement, the only neighborhood in wich codons similarities correspond to the similarities in chemical structures and properties (see Section 4.2.1 and footnote 14).
30 Only after these considerations and refinements can we continue with the analysis of specific chemical complementarity, valid for the standard genetic code. Namely, the nuancing and balancing of chemical structures (and properties) through complementarity becomes obvious only if we in the first and second columns (of the GCT) go downward, and in the last and the first to last M upward. By this act, there is an immediate obviousness that the "neighborhood logic" of codons and correspondent AAs, valid for columns, extends also to rows. Even more than the neighborhood logic also applies in diagonal connections (Comment 11) Comment 11. The presentation of chemical similarities ("connections") in columns, rows, and diagonals completely correspond to the presentation of chemical similarities between the chemical elements, as Mendeleev drew in the PSE Table just a year after the discovery of PSE (1870 vs 1869), drawing not only periods and groups, but also diagonal connections [Photocopy %JJJ 3L ' 4::8 3 )64' > :c, p. 341).]
The examples as follows. Phe and Leu are (through complementarity logic) in connection with Gly, as we said above. However, according to the logic of the neighborhood, they are at the same time in connection with Ser and Pro; Leu even more with Thr. In Phe the bonding between the head and the body was achieved on a methane and isobutane pattern; In Leu, the body itself is that pattern. In the Ser we have the first possible derivation of the methan pattern by the introduction of the hydroxyl group. [This introduction of a hydroxyl group chemically can also be understood as the substitution of one hydrogen atom in the CH3 group of Ala.] A "step further" is realized in the Thr M it is formed by replacing one hydrogen atom in the CH2 group of the Ser by a methyl group. Hence, in the Thr between the head and the body is not H-C-H, but H-C-CH3 group. [This also explains the similarity of Thr, both with Ala and with Pro, what corresponds to their positions in the GCT.] In the Pro, the side chain consists of three methylene (CH2) groups, analogous (in its commonality, as a sequence of three such groups) with the structure of the isopropyl group. AO... just as a MCH2M group in a molecule is called a / P 3M ' > ?' 778 By respecting the principle of self-similarity, the side chain of Pro can be chemically "read" as a pyrrolidine, which is a non-aromatic analogue of pyrrole (VI in Figure 1).]
31 Box 2. The deviant genetic codes At that very first time (from 1966 until 1979) the Genetic Code Table was considered to be the Table of a universal genetic code. However, later with the discovery of alternative genetic codes, the Table was renamed in GCT of standard genetic code. The universality of the genetic code was first challenged in 1979, when mammalian mitochondria were found to use a code that deviated somewhat from the "universal" (Barrell et al., 1979; Attardi, 1985). Our opinion about "deviant codes" we have expressed in one of the previous works 3 )64, 2004), and we still think the same today, that they represent only a "degree of freedom" in deviation from the standard one. [Knight at al., 2001, p. 49: "The genetic code # D ' QR code emerged before the last universal ancestor; subsequently, this code diverged in numerous nuclear and organelle lineages"; Weaver, 2012, pp. 568-569: "These deviant codes are still closely related to the standard one from which they probably evolved".]
After phenylanaline, going from the left to the right in the rows, we find a very characteristic case of simultaneous linear and diagonal neighborhoods between the three remaining aromatic AAs: Tyr, His and Trp. After phenylanaline and leucine in the first column, isoleucine and methionine come in a complementarity relationship with arginine. In other words, when we go upstairs, in the last column, we encounter the arginine for which we saw that in the side chain it possesses a structural motif analogous to the structural motif of the isolpropyl group (II and III in Figure 2). The same motif we find in Val at the bottom of the first column, as also, mutatis mutandis, a motif of iso-propyl and iso-butyl group branching we find at the top of the last column (Cys, Trp). With this movement M downward, upward, lateral and diagonal, we have the resolution for the position of the serine, the explanation of how it is located in two distant and chemically different places in the GCT. Serine is at the beginning of the second column as the diagonal adjacent to leucine. On the other hand, serine is adjacent to arginine, and arginine is located at the second end of the Leu-Arg complementarity line ("1-14"). It becomes clear that in the same serine column there is cysteine, which is its chalcogenic analog (both elements, oxygen and sulfur are in VI group of PSE).
4.3.2. Dissimilarity in similarity
At the beginning, as well as at the end of GCT, we have relations between two and two AAs, with the minimum change: Phe-Tyr and Gly-Ala, respectively, as explained in the Scenario in Chapter 2. We have the complementarity of two and two AAs, but not through the addition operation, but through the subtraction on the Binary tree (15 - 7 = 8 and 8 - 0 = 8). Tyrosine as the derivative of Phenilalanine, like alanine as derivative of glycinne, both represent the minimal possible change (as explained in the scenario, in Section 2.2.1 (Paragraph in front of Comment 2). Hence, this is complementarity by dissimilarity in similarity. [Notice that this complementarity expresses relationships between the first and third, as well as between the second and fourth columns in the GCT.
32 The cases that follow, listed simultaneously, in the upper and lower parts of Table 3 can be analyzed in the same or similar manner, as to "chemical eyes" are directly apparent. But this remains for some other occasions if it appears that there is of an interest of the scientific public.] The specificity of the chemical complementarity, which we are talking about here, is expressed as complementarity of the nucleotide doublets as well as the corresponding amino acids on each two vertices on the Boolean hypercube (B4) whose binary values give the sum 1111 in the binary record (corresponding to the number 15 in decimal one).
4.4. The cipher in relation to particles balance Table 4 directly corresponds with Rumer's Table (Table 2.1), in the sense that the classification of dublets is given into one-meaning (left) and two-meaning ones (right). On the left side, each four-codon family encodes one AA and the right one for two. [The exception is the family in the 7th position that is really two-meaning, but encodes one AA and termination signal.] In Table 4, nucleotide doublets are given in the order of the binary tree of the genetic code, while in the RumerR Table they are classified from the aspect of the number of hydrogen bonds: two quartets in the upper and two in the lower half of the Table; two upper with different (4 and 6), and two lower with the same number of hydrogen bonds (5 and 5). And these different classifications, with such grait differences in the type of organizational arrangements, were followed by nuancing and balancing in the number of atoms (in a set of 23 AAs): in the upper and lower half of the two arrangements, they differ by ± 0 (left) and ± 1 (right). [30 & 36 versus 30 & 36 and 89 & 84 versus 88 & 85 with a crossing.] The classification into one-meaning and two-meaning four-codon families, that is, four-codon and non-four-codon AAs, was followed by already well-known balance in the number of nucleons mediated by Pythagorean triples (Shcherbak, 1994) (Score 1110 vs 1110 at the bottom of Table 4). Moreover, by balancing and nuancing, the correspondence of this classification with Py-Pu distinctions in the standard GCT was established in a set of 61 of AAs. [See the result (330-66 = 264) vs 330 at the bottom of Table 2.1; cf. with result (330-66 = 264) vs 330 in ( )64, 2004, Table 3a, p.224); see also here in Survey 2 on the positions 8 & 10.] It is obvious that all these balancings and shadings are at the same time in the function of harmonization with the principles of similarity and self- similarity.
33 5. THE POLARITY IN RELATION TO THE CIPHER KEY
In the last paragraph of the Introduction of this chapter, we were told that we are starting from the hypothesis, according to which the chemical affinity between codon and amino acids in the genetic code is mediated by a specific cipher and its key. In the subsequent sections, arguments are given for making the cipher specific chemism in the form of a specific chemical complementarity; and that the cipher's key represents a sequence of codon positions and correspondent amino acids on the binary-ode tree of the genetic code ( )64, 1988, Fig. 31 on page 120; 1998, Fig. 1 on page 284). The following four Tables (5.1, 5.2 & 6.1, 6.2) contain the result, the final result, which supports that evidence. What is common in these tables is that it is directly apparent15 that the key of cipher splits AAs into two groups: a group in which the AAs are strictly differentiated into polar and nonpolar, and another group, in which this is not the case. What are different are changes through balancing and nuancing, expressed in changes in the number of molecules and atoms.
5.1. The final result The division of amino acids into two groups, in Table 5.1, is completely symmetrical. It is divided into the outer and inner space, each of four columns. Nuancing and balancing we find in interactions of Table 2.1, Table 4 and Table 5.1 are expressed by a change of ± 0 and ± 1 in the number of atoms by groups ("blocks") of AAs, with a crossover.16 The change is such that it also reflects the cyclicity of the system: in Table 4, the positions on the cipher key, from 10 to 14, participate in the balance, while in Table 5.1 we have a "block" movement for exactly one block. The block "1-14" appears to be out of the "game" and the block "6-9" comes in, so that now the positions 9 to 13, as an internal space, come to participate in the balance. In one cyclic movement, the block "1- 14" from the upper part of the Table is moved to the bottom; that is, together with the lower part makes the outer space. This is enabled by the same number of atoms of AAs H & Q and S & R (both times 22 atoms in their side chains); as well as by the difference in the number of atoms L-13 vs T-8 (for 5 atoms), whereby the previous result of "30" becomes as a new result 36-1, corresponded with the result 36, with an balance change for ± 1. [The changes for the ± 0 and ± 1 listed here, as in the previous sections, correspond with the same such changes (nuancing and balances) and in other systems and/or arrangements of the genetic code (Section 5.2)] In the dark spaces of Table 5.1, the separation into polar and nonpolar AAs is complete. On the other hand, in the internal unshielded spaces, the separation is such that
15 "A picture is worth a thousand wordsP 16 Dzivo Gundulic (Giovanni Gondola), Dubrovnik poet (1589M1638): "Tko bi gori, eto je doli, a tko doli gori ustaje" ("Who was up, now is down / and who was down, now is up.")
34 on one side we have a little polar AAs, and on the other side, the very polar. There is only one case where one opposite to the other we have non-polar AAs: a highly non-polar valine versus far less polar cysteine; and to make the separation more complete, tryptophan is added. The eight amino acids, which correspond to eight "one-meaning" nucleotide doublets (AAs in the upper part of Table 5.1), make sense to name - the single amino acids. In such a case, tyrosine joins them because it is also the only AA in the four-codon UA space (Table 5.2).17 As we see, this small change, this "tongue on the scale" causes new sophisticated shading and balancing, making "jumps" from one system to another, from one arrangement to another. The changing of Table 5.1 in Table 6.1 is that purine encoding AAs are excluded, those that are encoded by codons that have purine in the third position. The result (number of atoms) in the upper part of the two Tables has not changed since the same AAs remained. But in the lower part it is. Instead of 88, now we have 44 (which is again a symmetry and balance!). On the other hand, instead of 85, we now have 31, correspondent with the result 31 in the upper part. The difference 85 - 31 = 54 is in correspondence with the result of 44 atoms. The whole thing becomes clear when we compare Table 5.1 with Table 6.2 where pyrimidine encoding AAs are excluded: we have a balance in one crossing: 54 vs 44 and 41 vs 31.
Table 6.1. The horizontal CIS display into one-meaning and two-meaning nucleotide doublets and corresponding amino acids (III)
L S P V T A R G +2.8 -0.8 -1.6 +4.2 -0.7 +1.8 -4.5 -0.4
-0.8 -3.5 -3.5 +2.5 -3.2 -1.3 +4.5 +2.8 S D N C H Y I F
F.R. LARG = 35 SPVT = 31 III SYIF = 44 DNCH = 31
It is all the same as in Table 5.1, except that purine-coding amino acids are excluded in the bottom row.
17 In this definition, it is only important that the doublet UA can encode only one AA, irrespective of the fact that it can encode also the termination signal. 35 Table 6.2. The horizontal CIS display into one-meaning and two-meaning nucleotide doublets and corresponding amino acids (IV)
L S P V T A R G +2.8 -0.8 -1.6 +4.2 -0.7 +1.8 -4.5 -0.4
-4.5 -3.5 -3.9 -0.9 -3.5 ±0.0 +1.9 +3.8 REKW Q ʿ M L
F.R. LARG = 35 SPVT = 31 IV RML = 41 EKWQ = 54
(44 + 10 = 54) (31 + 10 = 41)
It is all the same as in Table 5.1, except that pyrimidine-coding amino acids are excluded in the bottom row.
The differences from the aspect of the inclusion / exclusion of the block of Py / Pu amino acids have also been shown to be significant in other arrangements of the Genetic Code. So, this was demonstrated in the case of the classification of AAs into two classes, handled by two classes of enzymes of aminoacyl- + / 3" &4 )64' >>6, 2006, 2007) (Box 3), and also in the case of the analysis of "p-adic model of ... genetic code " (Dragovich et al., 2006, 2010, 2017).
*
In Table 5.1, it should be noted that at the main "block" of the change, at positions "1- 14", there are amino acids L, S, R, precisely those that are doubled in the set of "23" AAs. Only with their duplication (Comment 12) is it easier to nuance and balance chemical properties, which could be a kind of "intelligent design" (Box 4). Now, the meaning of their duplication is seen: non-polar Leu vs polar Ser & Arg. In sequence "0- 15", we again have Le to "help" in establishing a relationship between non-polar Phe and polar Gly. The same applies to Arg, which, as polar, reappears in sequence "4-11" versus non-polar Ile and Met; Finally, Ser is also in the sequence of positions "2-13" in order to be, as slightly polarized against the high polar Asp and Glu. [For relations of the final result with some previous ones, see in Section 5.2.]
Comment 12. In the standard GCT the amino acids L, S, R appear twice each; in an extremely nuance reading of the standard GCT the I # 3 )64' 2007, Table 7; Wohlin, 2015, Table 2); within a "doublet-triplet" arrangement of protein AAs where reveals a connection between AAs and their biosynthetic precursors, the four AAs from non-alanine stereochemical typ 3$'H'%'J8 # 30/ )64 *)4' 4478
36 In Tables 5.1 & 5.2; then 6.1 & 6.2, we showed the nuancing and balancing in the number of atoms in the distinction of AAs to "single AAs" and "double AAs". In Tables 7 & 8, however, we showed nuancing and balancing in the "natural environment" of encoding amino acids, in the genetic code, from the aspect of as many as seven different parameters. These are the same two Tables that we published in one of the previous papers, but now more sophisticated, indicating the specific values for the hydropathy index.
*
The divisions, given by one and the same hatching within the six Tables (5.1 & 5.2; then 6.1 & 6.2, and in Tables 7 & 8), show that over the same division there is a specific proof of the correctness of the hypothesis about the necessity of distinguishing the genetic code on the code, the cipher and the key of the cipher. In a certain way, one confirms the other: the values of the hydropathy index confirm the validity of the cipher key, and the cipher key confirms the validity of the experimentally determined values of the hydropathy index (Kyte & Doolitle, 1982).
Box 3. Pyrimdine/purine distinctions in relation to two classes of AAs `A further classification is also a proof for the wholeness within a holistic system of genetic code, the classification in relation to the base type in third position of the belonging codon. Thus, in class II there are AAs whose codons do not possess purine in third position (first subclass): N, D, F, H ... with 40 atoms within their side chains; then AAs whose codons possess purine in third position (second subclass): K, P, A, S, T, G ... with 40+01 atoms. On the other hand in class I there are AAs whose codons possess either pyrimidine or purine in third codon position (first subclass): V, L, R, also with 40 atoms within their side chains; then AAs whose codons possess only purine in third codon position (second subclass): M, Q, E, W with 40+10 atoms; as a third subclass there are AAs whose codons possess only pyrimidine in third codon position: C & Y with 40-20 atoms within their side chains. Out of the classification within class I there is isoleucine which belongs to the first subclass within standard genetic code and to the thi # P (Damjanovi4 )64' >>6' 6 :; 2006 in arXiv:q-bio/0611033 [q-bio.OT], p.12; 2007, p. 121).
37 Table 7. The AAs sequence taken from GCT as well as from binary-code tree of Genetic Code (Rako6ev4, 1998a; 2004)
F L I M V Y H Q N K +2.8 +3.8 +4.5 +1.9 +4.2 -1.3 -3.2 -3.5 -3.5 -3.9
-0.8 -1.6 -0.7 +1.8 +2.5 -0.9 -4.5 -0.4 -3.5 -3.5 S P T A C W R G E D
After AAs encoded by middle "U" codons come AAs encoded by middle "A" codons; then follow AAs encoded by middle "G" and "C" in a cyclic organized system. The system can be seeing also as a sequence of the pairs (F-S, L-P, etc.). The sign "+" and "M" for nonpolar and polar AAs, respectively (after hydropathy index) (Rako64' >>9' able 7, p. 228).
Table 8. The results of calculations from data given in Table 7
AN MM NN-T NN-1 PN IN CN
Odd 102-1 1369-1 1513 627-1 343-1 210-1 203+1 Even 102+1 1369+1 1503 628+1 343+1 211+1 202-1
The designations: AN-the number of atoms within AA side chain; MM M the molecule mass of AA molecule; NN-T M the total nucleon number within AA side chain; NN-1M the nucleon number within first nuclide; PN M the number of protons; INM the number of isotopes (nuclides); CN M the number of conformations, as in Popov (1989, Table 8, p. 88). The sums are given for AAs pairs in odd (bold) as well as in even positions within the system in Table 7. For example, within five AAs pairs [(FMS), (IMT), (VMC), (HMY), (NME)], existing in odd positions, there are 10 AAs molecules with molecules mass of 1368 units and with atom number of 101 atoms, etc., as it is presented in this table. The balances are self-evident.
38 Box 4. The Spontaneous Intelligent Design Castro-Chavez, 2010, p. 718: "We can conclude that the genetic code is an intelligent design that maximizes variation while minimizing harmful mutations." )64' > ? > 6' 5= OM G propose a hypothesis (for further researches) that here, there really is a kind of intelligent design; not the original intelligent design, dealing with the question M intelligent design or evolution (Pullen, 2005), which is rightly criticized by F.S. Collins (2006). Here, there could be such an intelligent design, w # O0 J " P 30HJ"8 consistent with that design which was presented by F. Castro-Chavez (2010), and is also in accordance with the Darwinism. ... Actually, it can be expected that the hypothetical SPID, contai G' # 3 O8 0HJ"' / ' "#P ) OG 0 P 3"#' 5648' # # / ) 3 )64' 449C www.rakocevcode.rs). [In the case of the statement that spontaneity and intelligent design / ' ) I = R / OBP
5.2. The final result in relation to some previous ones In one of the previous papers 3 )64' 44:' 7968 we have shown that the distribution of amino acids within the "Codon path cube" (Swanson, 1984) is followed with a strict distinctiion into inner and outer space. Within inner space there are AAs handled by class I of enzymes aminoacyl-tRNA synthetases, whereas in outer one the AAs handled by class II. [The only one exception is arginine, handled by class I, which is entered with two its codons, into the space of amino acids handled by class II synthetases.] Within the two and two sub-spaces, in both spaces, the differences in the number of atoms in AAs (in their side chains) are also ±0 i ±1 (in relation to the mean value):
I: [(M11+I39+V40+L78 = 168±0); (Y30+Q22+E20+R68+C10+W18 = 168±0)] (1)
II+R34: [(T32+A16+P32+S20 +F28=129-1); (N16+D14+H22+K30+S10+G04+R34=129+1)] (2) In another paper 3 )64' > ' 5?5 59>8 we have shown that the external (o) and inner (i) space of the standard GCT is in relation to the space of nonpolar (n) and polar (p) AAs, respectively (also measured by the index of hydropathy), through the distinction in the number of atoms in AAs (in their whole molecules) which amount is exactly ±1:
(n) [(V76+M20+I66+A52+L132+F46+C28=420]; (o) [(V76+M20+I66+A52+Y48+R104+W27+C28= 421)] (3) (p) [(G40+K48+N34+P68+Y48+R156+W27+E38+D32+T68+S84+Q40+H40 = 723] (i) [(G40+K48+N34+P68+L132+F46+E38+D32+T68+R52+S84+Q40+H40 = 722] (4)
39 In the same paper (Rako64' > ' D 7' 5? D :' 5??8 # presented the CIPS (Cyclic Invariant Periodic System)18 of protein AAs, the canonical AAs within the genetic code. Cyclicism exists there in the sense that in the middle of the system there are chalcogenic amino acids (ST-CM), and further, in cyclic circles, there are two by two AAs from the same subclass, ending with two aromatic AAs and two aromatic heterocyclic (FY-HW). The obtained cyclic rings split then into two superclasses: superclass I with AAs of lower chemical complexity (GP-VI + AL-KR) and Superclass II with AAs of more chemical complexity (ST-CM + DE-NQ + FY-HW). The number of atoms in two superclasses and two classes is well balanced with the difference of ± 0:
Class II: {[(G01+P08)+(A04+K15) = 28] + [(S05+T08)+(D07+N08)+(F14+H11) = 53]} = 81 Class I: {[(V10+ I13)+(L13+R17) = 53] + [(C05+M11)+(E10+Q11)+(Y15+W18) = 70]} = 123 (5)
Sup.cls I: {[(G01+P08)+(A04+K15) = 28] + [(V10+ I13)+(L13+R17) = 53]} = 81 Sup.cls II: {[(S05+T08)+(D07+N08)+(F14+H11)=53]+[(C05+M11)+(E10+Q11)+(Y15+W18) = 70]} = 123 (6) It is obvious that this is a "block" exchange between two classes and two superclasses ( )64, 2011b, Figure 7, p. 833).19
6. CONCLUDING REMARKS
The facts about a specific chemical complementarity of the constituents of the genetic code, given throughout this chapter provide evidence to support the hypothesis, given in the title of this chapter that the genetic code can be interpreted as the unity of the three entities: the code, the cipher of the code and the key of the cipher. Just only with this understanding, we can find, within the two main Genetic Code Tables (of the nucleotide doublets and nucleotide Triplets) the sophisticated nuancing and balancing in the properties of the constituents of GC, including the balance of the number of molecules, atoms, and nucleons. All this also confirms our hypothesis, given in one of the 3 )64' 2004) that the genetic code, from the beginning, in prebiotic conditions, was complete. Out of the millions of possible aggregations of molecules, the potential builders of GC, on "the card of life" played the one that potentially possessed all the chemical complementarities, we have exposed here.20
18 CIPS follows from determination of genetic code by Golden mean on the Binary- 3 )64' 1998, 2011b), which tree appears now as the key of the ciher of the Genetic code. 19 According to our prediction, these relationships will be, in the future, a referent system for protein structure researches. 20 "Each of that aggregations could (and must) have its own QR' / selected M the one that gained the characteristic of self-reproduction (by which, through trial, error and success it became everything); all other, not selected, could not have any chance ..., they became nothing" 3 )64' >>9' ?8
40 Our expectation is that this work could be joined to works that are on the way to solving open problems of existence and the essence of genetic code; such problems related to the search for answers to questions of origin and evolution of life; in particular, those works that open up new fronts of research in biology, extending the theme of genetic code on topics of The codes of life, or, more broadly, to The biological codes (Barbieri, Hofmeyr et al., 2008, 2018).
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44 Chapter 2
Analogies of Genetic and Chemical Code
Starting from the previously established hypothesis on the existence of "the coherence of @ 3 )64' 44 8' # # existence of essential analogies between the genetic and chemical code are presented, whereby the chemical code is derived from the original Mendeleev's work on the periodic system of chemical elements. When, namely, the 14 lanthanides are scheduled into 14 H / 3H0T8 3 )64' 44 8' to Mendeleev's original manuscript tables, then are revealed some very specific mathematical and chemical regularities within the PSE; among others, PSE appears to be a code M the chemical code M in an entity analogy with the terrestrial genetic code. These regularities correspond to the mathematical regularities within the genetic code. Between others relations, it appears also a correspondance between the distribution of codons in the Genetic Code Table and the distribution of chemical elements in the PSE with respect to their even/odd parity and stability/instability of the isotopes. Based on the significant mathematical expressions it is showed a new essence of coding formalism in natural code as such: it cannot be reduced only to two alphabets (which are consequences, and not the cause); a natural code is such an essence that can be represented by an appropriate mathematical expression, which contains a sequence from the series of natural numbers within itself. From this follows that both are natural codes, the Genetic code as well as the Periodic system of chemical elements (PSE).
1. Preliminaries 1.1. Analogies with quantum physics V. Shcherbak showed that a specific classification of the protein amino acids (AAs), "canonical" amino acids within the genetic code (GC), reveals some arithmetical regularities and stated that "the physical nature of such a phenomenon is so far not clear" (Shcherbak, 1993). There is a similar statement of A. Verkhovod in his work on the same subject and published a few months later: "The nature of these mechanisms is presently unknown" (Verkhovod, 1994). In the second part of his rsearches Shcherbak (1994) "seeks to identify a trend of possible physical interpretation of the new code properties", and reveals that in the question are "analogies with quantum physics." By this, the Shcherbak's classification refers to the splitting into four-codon and non-four-codon AAs, based on the original work of Rumer (Rumer, 1966) (Table A.1 in Appendix A). If so, then there are analogies in the next sense: 01 x 111 of electrons in the half-filled state of
45 the atom orbitals, or 01 x 222 in the full-filled state, according to Hund's rule, versus 10 x 111 nucleons in 15 side chains as well as in 15 "heads", i.e. amino acid functional groups (after Shcherbak: "standard boxes") of 15 non-four-codon AAs, what equals 10 x 222 in 15 whole molecules; at the same time there are 01 x 333 nucleons in 8 side chains of 8 four-codon AAs (for details see: Shcherbak, 1994)1 Considering that 01 x 333 can be "read" as 01 x 3^2 PQ, then we find that in 8 "heads" of 8 four-codon AAs there are 01 x 4^2 PQ, or in total 01 x 5^2 PQ of nucleons, what is again an analogy with quantum physics, as well as with two Mendeleev principles at the same time (see Remarks 1-3).
Remark 1. Whether it is a move for one electron in filling of the orbital, or for a single numbering unit in filling the positions in the number record of the number of nucleons, both have correspondence with two key Mendelevian principles: with the principle of continuity and the principle of minimum change. In addition to this, it should be noted that this is a special case of a minimum change M a change for a unit;2 and there is also an analogy with the changes on the Gray code model of the genetic code,3 as well as on the GC binary tree that follows from the Gray Code.4 Remark 2. The uniqueness of the number 037 in the Shcherbak's pattern "Prime Quantum 037" (PQ) follows from the uniqueness of a specific sequence, generated from the series of 5 natural numbers: (134, 257, 3710, 4913, ...) , for which two Mendeleev's principles also are valid. [The validity through the sequence 1, 2, 3, 4, etc., in front position, and through the sequence 3, 5, 7, 9, etc., in ending position.] Remark 3. The chemical and formal bond between four-codon and non-four-codon amino acids is described in the next Section. However, one specific formal-mathematical relation, in a certain way hidden,6 we present here: (10 x 3^1 PQ) nucleons within 15 side chains of 15 non- four-codon AAs versus (01 x 3^2 PQ) nucleons within 8 side chains of 8 four-codon AAs. It should be noted here that the "15" encoded entities are actually 16 encoded entities because
1 Shcherbak (1994, p. 475): "The cause of the regularity lies in the properties of three-digit number notations, multiples of 37 in the decimal additive-position system. The notations of the sums, the nucleon number of the amin U >:9 H V >?: G / pattern of the general regularity of Table 1" (here: Table A.2). 2 @0 / G U/ 3 8 the unit change law" (R)66' 449' ?78 3 @G / # & / A $/ B / @ 30# ' 459' 558 4 "The binary-code tree [of the GC] corresponds per se with the Gray code model of the genetic code (Fig. 1 in Swanson, 1984 p. 188)" (Ra)64' 445' 5?8 5 Is it just as a curiosity, or more than that, this can not be known, but the fact is that the number of verses in the 100 songs of Divine Comedy of Dante Aligierie corresponds with this sequence: the number of verses in any of the 100 songs is so large that the sum of the digits of the numeral record is 4, 7, 10 or 13, without any exception. 6 We say "hidden relation," since from the aspect of valid paradigms in the current science it is not expected that "quantization" in positions in the decimal numbering records could in any way be analogous to quantization in quantum physics. 46 the 16th entity joins a "stop" signal. In this case, the ratio of four-codon and non-four- codon situations is in a strictly symmetrical manner of view, 8: 16 = 1: 2 (cf. Survey 1 and Table A.1).
As far as Verkhovod's work is concerned, he showed that the number of nucleons in the 23 amino acid molecules in the standard GCT directly corresponds with the sequence of the series of natural numbers (in decimal numbering system) in a specific way, through the "game" of the visible and invisible image in the "mirror": 456/789 and 987/654. Namely, in amino acid side chains, within (1th+4th ) rows, as well as within (2nd+3rd) columns there are 654 nucleons in both area; vice versa, within (1th+4th) columns, as well as within (2nd+3rd) rows there are 789 nucleons in each of two area (Figure 2 in Verkhovod, 1994).
1.2. Agreement-disagreement principle As we can see in above presented classification of AAs appears a specific unity of chemistry and mathematical formalism, which in this chapter will be analyzed in details, starting from both works, Rumer's and Shcherbak's. In his first work on the genetic code, Rumer (1966) shows that four variations of nucleotide doublet CG (CC, GG, CG, GC) with 6 hydrogen bonds (higher rank!) encode four-codon AAs (higher rank!), while four variations of UA (UU, AA, UA, AU), each with 4 hydrogen bonds (lower rank!) encode non-four-codon AAs (also lower rank!). However, the situation is more "complicated" in the coding with nucleotide doublets that have 5 hydrogen bonds each. There is a characteristic "crossing", but also in relation to a strict chemical distinction. Thus, nucleotide doublets that in the second position possess a more complex nucleotide (CA & UG), i.e. the nucleotides of higher rank, encode non-four-codon AAs, what is per se a situation of lower rank. On the other hand, nucleotide doublets that in the second position possess a less complex nucleotide (AC & GU), i.e. the nucleotides of lower rank, encode four-codon AAs, what is the situation of higher rank. Finally, pyrimidine doublets (UC & CU) (lower rank!) encode four-codon AAs (higher rank!), while purine doubles (GA & AG) (higher rank!) encode non- four-codon AAs (lower rank!). As we see, the state of things is such that the disagreement in the number of hydrogen bonds (6 & 4) is accompanied by an agreement in the rank of the complexity of the molecules and / or the complexity of the coding process; vice versa, the agreement in the number of hydrogen bonds (5 & 5) is accompanied by disagreement in the ranking. In a certain way, here we indeed have an analogy with quantum physics, more precisely with Heisenberg's uncertainty principle (in the form of an agreement/disagreement principle).
47 1.3. Some other mathematical formalisms Independently of Shchrbak and Verkhovod, other authors, during the last decades, have also presented other different arrangements in which chemical distinctions are accompanied by some kinds of mathematical formalism. So, R. Swanson has shown that the genetic code can be reduced into a strict mathematical-formal model, determined with the "Gray code binary symbols for numbers 0-63", and also reduced into the "Codon path cube" in which "all 64 codons are displayed", and where "the three edges of the cubes represent the three positions in a codon" (Swanson, 1984). By this, in both cases the principles of minimum change and continuity are valid, the same principles which are valid for the arrangement of chemical elements in the periodic system of D.I. Mendeleev. R. Swanson, however, showed another more important thing. The "Codon ring" in the form of a Gray code model, which we have in GC (input!) and the "Mutation ring" of the amino acids, which is the result of the evolution of proteins in the terrestrial organisms (output!) are more than very similar M mutatis mutandis they are the same. With this insight, she rightly states that these findings "provides a new standpoint for addressing questions of selection vs random drift in the evolution of the code." The additional reason for this reexamination is the fact that in both rings (Codon and Mutation ring) AAs are classified into small/large and inner/outer, in terms of their position in proteins, with directly visible certainty/uncertainty of these positions, which is again an analogy with quantum physics, i.e. with Heisenberg's uncertainty principle. Using the same rules7 that were applyed to the generation of the Gray Code model of GC, it was possible to obtain a six-bit binary tree of GC, and show that the GC is determined by the Golden mean ( )64, 1998b). On the other hand, if the splitting of the amino acids into two (by enzymes aminoacyl-tRNA synthetases handled) groups, instead in the standard Genetic Code Table (GCT) (Wetzel, 1995), is made in Codon path cube, the significant regularities are noted M the separating of two classes of AAs with only one exception ( )64, 1997a) (see Table B.1 in Appendix B).
Remark 4. The six-bit binary tree is specific, in addition to everything else, in following: only at such a binary tree the sums of numbers within two inner branches (two octets) corresponds to the first pair of friendly numbers (220 + 284 = 504); and each two adjacent branches give the same sum (504); all together, a kind of logical square is realized: (0) 220 + 284 = 504; (1) 156 + 348 = 504; (2) 92 + 412 = 504; (3) 28 + 476 = 504. On the other hand, the sum of the numbers of the first quartet is 6, the first octet 28; the sum of numbers on the left half (which coincides with the first half of the GCT), from 0 to 31, is 496, which in turn is the realization of the first three perfect numbers (6, 28, 496). If, however, we count all numbers 0-63, then go back (cyclic!), where 0 (zero)
7 "The most important characteristic of a codon is whether it has a purine (most significant bit 1) or a pyrimidine (most significant bit 0) at the middle G@' 30# ' 459' 558 48 becomes to be 127, then the sum of all numbers within the sequence 0-127 equals 8128, which is actually the fourth perfect number. After these insights, it is clearer the analogy between the binary records of 64 codons in the genetic code and 64 hexagrams in the Chinese book I Ching, at least three thousand years old; the records, on the six-bit binary- code tree in both cases (Stent, 474C 0 ' 46 C )64' 4498 (www.rakocevcode.rs).
A year later, after R. Swanson's published work, an article more about specific mathematical formalism, accompanied by the chemical distinctions of AAs according to the number of hydrogen atoms in their molecules has been published (Sukhodolets, 1985). Such a distinction corresponds with the characteristic sequence from a series of natural numbers: 5, (6), 7, 8, 9, 10, 11, 12, 13, 14 of hydrogen atoms (Appendix C). On all the said and some other similar investigations of mathematical regularities of the GC one can see the following works: (Sukhodolets, 1985; Leunissen and De Jong, 1986; Koruga, 1992; Madox, 1992, 1994; Shcherbak, 1993-2005C " &4' 445- 2006; Qiu and Zhu, 2000; Yang, 2004; Dragovich, 2006-2012; Negadi, 2009-2014; Castro-Chavez, 2010, 2011; -4, 2011; Petoukhov, 2014; 2016; Wohlin, 2015; )64' 455-2014).
2. New insights 2.1. The problem of lanthanides arrangement in the PSE The reason why it has not been previously noted the possible analogies between mathematics of the GC and mathematics of the PSE, lies in the fact that all 14 lanthanides are placed in the third group of PSE, i.e. at the same position where is the Lanthanum. However, this is in disagreement with Mendeleev's approach that every element in the PSE should have its own position. Namely, the book of B.M. Kedrov (Kedrov, 1977, p. 188, Table 16) contains a variant of Mendeleev's Periodic Table, in which Mendeleev did not formally indicated the groups, but it is evident that each element occupies one position. In that Table, lanthanum is located in the third group, Cerium in the fourth group and so consequently all other 13 elements, although two elements (Pm & Lu) were not known in Mendeleev's time. (Some Mendeleev's manuscript Tables can be seen in the author website, www.rakocevcode. rs.)8 This problem with lanthanides positions in the PSE is still actual, because recently IUPAC (International Union of Pure and Applied Chemistry) has launched a new research project which should determine whether lanthanum with atomic number 57, or
8 There was, however, an attempt to "integrate" the lanthanides into the Periodic System, by Charles Janet (1849-1932), so each of them was in a separate group. Unfortunately, it was only in our time recognized that Charles Janet was "unrecognized genius of the Periodic System" (Stewart, 2010).
49 the last lanthanide, Lutetium with atomic number 71, should be written in the formal Table of PSE (see Appendix D). Following Mendeleev's methodology, it was possible to show that the 14 lanthanides require exposition into 14 groups of the PSE. Then together with zeroth group, there are 15 groups ( )64' 44 8 (see Table 1 in this chapter). If we have such an arrangement, then it is easy to recognize not only arithmetical but also some algebraic regularities in the PSE. In a #) 3 )64' 44 8 # hypothesis that the PSE of the short period groups corresponds to the Boolean cube as well as the PSE of the long period groups to the Boolean hypercube;9 the role of the 16th group in such a case (in a cyclic ordering) plays either zeroth group or the first one. In fact we mainly pay attention to this chemical code, because it is an analog of the genetic code.10 [Mendeleev also entered the elements of the first group M copper, silver and gold M twice, at the beginning and at the end of the PSE (Kedrov, 1977, p. 128, photocopy XII).]11
2.2. Specific arithmetical patterns Table 1 (in relation to Survey 2) shows that for [(s & p), d, f] elements,12 to the stability/instability border in PSE (to the Po-84), we have 8 times the pattern 5-3-1; then 2 times the pattern 0-3-1 and 4 times the pattern 0-0-1. All together 9-4-1 elements: 9 elements 8 times; 4 elements 2 times and 1 element 4 times; the patterns 9-4-1 and 8-4-2 as unique and very specific mathematical expressions (Eq. 1).
9 "Such a surprisingly simple model at the same time represents the Logical-Informational and Geometrical-Homeomorphic-Topological model (LIGHT) of the cube-hypercube with an inscribed sphere- / @ 3 )64' 44 ' 8 10 "G ' / # G with / // / G@ 3 )64' 44 ' 8 11 In this chapter, we will deal with only the standard genetic code, with the 20 canonical (protein) amino acids, and all other variants of the genetic codes will be co Q R 3M' > ' 568-569). These codes, namely, do not change nothing on the fact, when it comes to the genetic code as "amino acid code" (Swanson, 1984), because they are represented in all variants with the same 20 protein amino acids. (The only exceptions are Pyrrolyisine and Selenocysteine, presented at some very few organisms.) [Knight at al., 2001, p. 49: "The genetic code evolved in two distinct phases. First, the QR tor; subsequently, this code diverged in numerous nuclear and organelle lineages".] 12 The chemical elements of s & p types we consider (within this chapter) as "intransitive elements", then the elements of d type as "the first transitive elements" and, finaly, the elemets of f type as "the second transitive elements".
50 Table 1. Periodic system of chemical elements with 14 lanthanides in 14 groups (Table 4.2 in: )64' 44 ' 5 = )64' 1997b).
(12 + 22 + 32 = 1+4+9) / (21+22+23 = 2+4+8) 3 3 B n2 = 14 B 2n = n1 n1 ...... (1) 14
(11 +21+31 = 1+2+3) / (11+12+13 = 1+1+1)
3 3 Bn1 = 6 B1n = n1 n1 ...... (2) 3
51 1. (m1, m2, m3); (m = 4) ...... (3) 2. (n1, n2, n3'G' 6); (n = 2) ["m" as GC alphabet; "n" as binary alphabet (0, 1), valid for the genetic code binary tree.13 3. mn = nm = 16 (m = 4); (n = 2) ["m^n" as number of doublets in Surv. 1; "n^m" as number of four-codon families (16) in Table 5 as well as on the genetic code binary tree.]
Table 1.1. The n-th powers (n = 0, 1, 2, 3) of the first three numbers
13 23 33 36 (-3) 39 31 32 33
12 22 32 14 (±0) 14 21 22 23
11 21 31 6 (+3) 3 11 12 13
10 20 30 3 (+3) 0 01 02 03
24 = 16 16 = 42
13 Through designated exponentiations from "m" follow nucleotide singlets and doublets in Survey 1, and triplets in Table 5, and from "n" the number of branches on the genetic code binary tree.
52 Table 1.2. The relations between two significant sequences of natural numbers in correspondences with the quantities of the genetic code constituents
[(29 = 512) (92 = 81)] Correspondences 02: Two letter alphabet, one letter words 3KF 8 0I "+ [(28 = 256) (82 = 64)] (+192) [3 x 64] 04: Two letter alphabet, two letter words 3K F K8 ( "+ [(27 = 128) (72 = 49)] (+79) 08: The 8 great codon families (YUN, YCN, YAN, YGN); (RUN, RCN, RAN, RGN) [(26 = 64) (62 = 36)] (+28) [4 x 7 = 28] [4 x 9 = 36] 16: The 8 small codon families [(UUN, CUN, 5 2 (+7) AUN, GUN); (UCN, CCN, ACN, GCN)]; [(2 = 32) (5 = 25)] [(UAN, CAN, AAN, GAN);(UGN, CGN, AGN, GGN)] [(24 = 16) (42 = 16)] (±0) 32: The 32 Y + 32 R codons
[(23 = 08) (32 = 09)] (-1) 64: The 64 codons
[(22 = 04) (22 = 04)] (±0)
[(21 = 02) (12 = 01)] (+1)
[(20 = 01) (02 = 00)] (+1)
53 Table 1.3. The solutions of significant linear equations (I)
... (a) (b) (c) (d) (e) x + y = 144 122 22 (72 & 72) ʶ 50 (00) 12 x - y = 100 (40) (18) 10
x + y = 100 82 18 (50 & 50) ʶ 32 (00) 10 x - y = 064 (32) (14) 8
x + y = 64 50 14 (32 & 32) ʶ 18 (00) 8 x - y = 36 (24) (10) 6
x + y = 36 26 10 (18 & 18) ʶ 08 (00) 6 4 x - y = 16 (16) (06)
x + y = 16 10 06 (08 & 08) ʶ 02 (00) 4 x - y = 04 (08) (02) 2
x + y = 04 02 02 (02 & 02) ʶ 00 (00) 2 x - y = 00 0
In first column there are the significant linear equations. At their right side are the squares of the numbers in (e). The other relations as follows; a = x, b = y and c = x and y with minimal changes of them in columns a and b, respectively; d = the difference between a and b.
54 Table 1.4. The solutions of significant linear equations (II)
... (a) (b) (c) (d) (e) x + y = 121 101 20 (61 & 60) ʶ 40 (01) 11 x - y = 081 (36) (16) 9
x + y = 81 65 16 (41 & 40) ʶ 24 (01) 9 x - y = 49 (28) (12) 7
x + y = 49 37 12 (25 & 24) ʶ 12 (01) 7 x - y = 25 (20) (08) 5
x + y = 25 17 08 (13 & 12) ʶ 04 (01) 5 x - y = 09 (12) (04) 3
x + y = 09 05 04 (05 & 04) ʶ 00 (01) 3 x - y = 01 1
In first column there are the significant linear equations. At their right side are the squares of the numbers in (e). The other relations as follows; a = x, b = y and c = x and y with minimal changes of them in columns a and b, respectively; d = the difference between a and b. [Notice that the case (x + y = 25), which we found in genetic code, as well as chemical code, corresponds to the best possible case in Table 1.5 (13 vs 13 and 12 vs 23 with a difference of 11 what is the key determinant of TMA system (Figure 1 in Chapter 3). Notice also that the numbers 12 & 13 are very unique in the series of natural numbers (cf. Table A.1 in Chapter 4.]
55 Table 1.5. The solutions of significant linear equations (III)
... (a) (b) (c) (d) (e) x + y = 144 122 22 (117 & 27) ʶ 05 (90) 12 x - y = 100 (40) 10
x + y = 100 82 18 (80 & 20) ʶ 02 (60) 10 x - y = 064 (32) 8
x + y = 64 50 14 (47 & 17) ʶ 03 (30) 8 x - y = 36 (24) 6
x + y = 36 26 10 (23 & 13) ʶ 03 (10) 6 x - y = 16 (16) 4
x + y = 16 10 06 (08 & 08) ʶ 02 (00) 4 x - y = 04 (08) 2
x + y = 04 02 02 (02 & 02) ʶ 00 (00) 2 x - y = 00 0
The Table follows from Table 1.3. Only in the case (x + y = 36) there is a correspondence with the principle of similarity and self-similarity (03, 13, 23), the best possible case, which we found in chemical element isotopy, as it is shown in Survey 2. In the column (d) is a search for the solutions which can be in relation to the said M the best possible.
56 Table 2. Periodic system of chemical elements with 6 groups
The expression in Eq. (1) is related to the number of chemical elements in Table 1. On the other hand, the expression (2) is related to triads, diads, monads, respectively, in Table 2 and Table 3 (in relation to Table 4). In addition one can notice that in reality to the Eq. (1) precedes Eq. (2) as a previous step in exponentiation, valid for the first three natural numbers. The expression in Eq. (3) shows the relationships within the genetic code (cf. Survey 1 and Table 5). As a special case is the set {m1, m2, m3} in an indirect correspondence with the set {21+22+23} in Eq. (1) what means the correspondence among genetic code and chemical code. [Eq. (1) corresponds with Table 1; Eq. (2) with Table 2 and Tables 3 & 4; Eq. (3) with Survey 1 and Table 5. Both Eq. (1) and Eq. (2) correspond to Tables 1.1 and 1.2.]
Remark 5. Table 1 is essentially a periodic system of short periods. As in the original works of Mendeleev, it can develop into a periodic system of long periods when it per se has a satus of a block-PSE14 as a set of adjacent groups: s-block, p-block, d-block and f- ) 3 )64' 44 ' 9? )64' 44:' 48 (www.rakocevcode.rs). Otherwise, the term "Block Periodic System" as well as "Block Periodic Table" appears to have been first used by Charles Janet, which is understandable
14 The block names (s, p, d, f) are derived from the spectroscopic notation for the associated atomic orbitals: sharp, principal, diffuse and fundamental. 57 (footnote 8), because talking about blocks makes sense only if lanthanides are distributed into 14 groups.
In Table 2 we actually have the correspondence with the mathematical expression in Eq. (2): 1 set of monads, 1 set of diads, 1 set of triads; at the same time: within the set of monads the isotope number relationships are realized through the singlets of chemical elements; within diads through doublets, and within triads through triplets. From the 1st to the 8th group, with sub-groups a, b, c, in the PSE (Table 1) are realized the elements correspondingly with the first member of the first mathematical expression in Eq. (1), in the form of (5+3+1= 9) elements: 5 intransitive elements (s or p), 3 first transitive elements (d) and 1 second transitive element (f). In the final (eighth) case, the zeroth group has the status of 8th group, with 5 intransitive elements. [The group of noble gases has zeroth group status in terms of chemical reactivity, and the status of eighth group in terms of filling the orbitals by electrons.] After the eighth group (with sub- groups b, c) are realized the elements into the ninth and tenth group, corresponding to the second member of the mathematical expression in Eq. (1), in the form of (3 + 1): 3 first transitive elements (d) and 1 second transitive element (f). Finally, after tenth group, followed the groups: XI, XII, XIII and XIV, each group with one "second transitive" element (f) correspondent with the third member of the mathematical expression, given in Eq. (1). [Notice that the first three periods are single, each with a single row; fourth and fifth are double, each with two rows and sixth period is threefold M it has three rows.] Within Table 2 all isotopes (indicated in the brackets) are naturally occurring (stable plus unstable primordial). The exception is P # Q R' as a result of radioactive decay of its isotopes existing in the nature in trace amounts, or synthesized: Pb-204, Pb-205, Pb-206, Bi-208, Bi-209. [This six-groups-PSE one can cf. with the first Mendeleev's Table of PSE, with 6 groups, March 13, 1869, in: Kedrov, 1977, p. 128, photocopy II; also in: Mendelejeff, 1869, reprinted in 1970.] Let us notice that the arrangement in Table 1 corresponds to the PSE, constituted of 14 groups and 6 periods, and the arrangement in Table 2 corresponds to the PSE constituted of 6 groups and 14 periods. These two PSEs are in correspondance with each other: odd elements are in odd groups, and even element in even groups M in both systems. 3E 9 < 9 = )64' 44 ' 5 < > = )64' 1997b.) (The Reference: )64' 44: e in website: www.rakocevcode.rs)
2.3. Quantum like relationships within nucleotides arrangements The expressions in Eq. (1-3) show the essence of coding / encoding in a natural code; it cannot be reduced only into two alphabets (which are consequences and not the cause). A natural code is such an essence that can be represented by an appropriate mathematical expression, which contains within itself a sequence from the series of natural numbers.
58 From this follows that both, the Genetic code as well as the Periodic system of chemical elements (PSE) are natural codes. Bearing all this in mind, the expression N= nk, valid for the genetic code, is "readable" in the following way: n = 4; k = 3, where 4 is the number of letters within the GC alphabet, 3 is the number of letters within a word, 2 is the root of the word, and 1 is the number of letters within one letter; altogether, 1, 2, 3, 4, as the sequence of the natural numbers; finely, N is the number of codons, which encode "letters" from another alphabet, i.e. molecules of AAs. As we see, both Mendeleevian principles are valid, which could not be the case, for example, for n = 5, when, in the case of validity of Mendeleevian principles, we have uncertainty about the root of the word M is it of two or three letters? The expressions in Eq. (3) show multiple relationships of nucleotide doublets and AAs in Survey 1. But in order to understand these relationships, it is necessary to analyze the relationships between nucleotide doublets arrangement in Survey 1 and arrangement in the original Rumer's Table (Table A. 1); also in order for better understand the sense of our new insight. By comparing two Tables, we find that both in the upper part contain nucleotides with 6 and 4 hydrogen bonds, respectively, that encode AAs, which have 119 atoms in their side chains. In the lower part there are nucleotide doublets with 5 and 5 hydrogen bonds, which encode AAs, which have 120 atoms in their side chains. It can therefore be said that here, in up/down direction, exists a balance with the difference for ±1 atom. However, significant differences exist from the aspect of the left/right organization (cf. Survey 1 and Survey A.1).
Remark 6. The number of atoms in the four molecules that build messenger RNA, corresponding to the GCT, is: U = 12, C = 13, A = 15, G = 16; in doublets: UG = 28 x 1 and CA = 28 x 1. When the nucleotide form is taken, then the number of atoms is: U = 34, C = 35, A = 37, G = 38; in doublets: UG = 36 x 2 and CA = 36 x 2. [Cf. (Rako64' 1997a); (Rako64' 44:= 0 6-29, pp. 62-63; www.rakocevcode.rs)]. The question arises: is this a curiosity and a coincidence, or is it a certain natural-code essence when the sums of the number of atoms per doublets correlate with the square of the first perfect number (6) and the first degree of the second perfect number (28)?
In the Rumer's Table (Table A.1) on the left side are nucleotide doubles that encode four-codon AAs and on the right side ones that encode non-four-codon AAs. In our Table (Survey 1), however, the nucleotides are mixed; from both types, some of them are on the left and some on the right in original Rumer's Table. Nevertheless, when it looks more carefully, then it can be seen that this is a strictly symmetrical arrangement, a type of mirror symmetry. To more complex "geometric figures" (higher rank!), on the left, correspond less complex "figures" (lower rank!) on the right side: [(UC-CU-GU / CC- GG) // (AU-UA / CA-GA-AG)]; and vice versa: [(UU-AA / UG) // (AC / GC-CG)]. But
59 that what is particularly interesting, and in some way very surprising, is the fact that all this "geometry" is accompanied by a "quantum" change for 1 unit in the second position of the decimal number record of the number of atoms: 110 versus 120 and 129 versus 119. (For details see Surveys A.1 and A.2). There is also an additional question M what is the sense of the existence of a link between the left and the right side of the Rumer's Table; the link, expressed through the mirror symmetry and the quantization, presented in Surveys 1 and Surveys A.1, A.2, A.3? The possible respond is: in all here presented arrangements on the scene is a nuance principle in polarity of the amino acid molecules. [For the same reason, some AAs in GCT as well as in the "doublet-triplet" arrangement have been duplicated.]15
Survey 1. Generation of 16 nucleotide doublets from four-letter alphabet e = 1 e = 2 an U C UU AA CC GG AU UA GC CG F,L N,K P G 59 I,M Y,ct A R 60 119 A G UC CU GU UG AC CA GA AG S L V C,W 51 T H,Q D,E S,R 69 120 110 129
22UUFL, 22AANK, 33CCP, 33GGG 22AUIM, 22UAYct, 33GCA, 33CGR
23UCS, 32CUL, 32GUV, 23UGCW 23ACT, 32CAHQ, 32GADE, 23AGSR
The generation of the 16 nucleotide doublets from four-letter genetic code alphabet, according to the expression m ^ n (m = 4; n = 2; e = exponent to alphabet "m"; an = number of atoms). If e = 1, then we have four singlets, the four bases M two pyrimidines (uracil, U; cytosine, C), with one ring in the molecule (chemically simpler), and two purine bases (adenine, A; glutamine, G), with two rings in the molecule (chemically more complex). If e = 2, then we have 16 nucleotide doublets. If these doublets are arranged in a chemical hierarchy (first simpler with two hydrogen bonds: UU, AA; then more complex, with three hydrogen bonds: CC, GG; in further steps come their variations, and first row (upper row) is made. These doublets are the same as in upper area of Rumer's Table (Table A.1). Then follows the generation of the lower row of nucleotide doublets, with the idea to present possible mirror symmetry. The result is the discovery of a "hidden" link between two sides of Rumer's Table (Table A.1): of the left side with four-
15 In the standard GCT the amino acids L, S, R appear twice each; in an extremely nuance reading of the standard GCT the isoleucine appears also twice 3 )64' >>:b, Table 7; Wohlin, 2015, Table 2); within the "doublet-triplet" arrangement of protein AAs where reveals a link between AAs and their biosynthetic precursors, the four AAs from non-alanine stereochemical type (G,P,V,I) appear twice (Survey 1 and Table 1 in Rako64 *)4' 4478
60 codon AAs and of the right side with non-four-codon AAs. As it is obvious, the "mirror picture" is expressed through specific "quanta" of the number of atoms in the side chains of AAs as well as through "quanta" of the sets of amino acid molecules (cf. Survey A.1 & A.2).
Survey 2. Distribution of isotopes within the PSE (1) Group I:
{[3Li (2+0)], [11Na (1)] & [19K (2+1)], [37Rb (1+1)], [55Cs (1)]} {[29Cu (2+0)], [47Ag (2+0)], [79Au (1)]} {[69Tm (1)]}
(2) Group II:
{[4Be (1)], [12Mg (3+0)] & [20Ca (5+1)], [38Sr (4+0)], [56Ba (6+1)]}
{[30Zn (5+0)], [48Cd (6+2)], [80Hg (7+0)]} {[70Yb (7+0)]}
(3) Group III:
{[5B (2+0)], [13Al (1)] & [31Ga (2+0)], [49In (1+1)], [81Tl (2+0)]} {[21Sc (1)], [39Y (1)], [57La (1+1)]} {[71Lu (1+1)]}
(4) Group IV:
{[6C (2+0)], [14Si (3+0)] & [32Ge (4+1)], [50Sn (9+1)], [82Pb (4+0)]} {[22Ti (5+0)], [40Zr (4+1)], [72Hf (5+1)]} {[58Ce (4+0)]}
(5) Group V:
{[7N (2+0)], [15P (1)] & [33As (1)], [51Sb (2+0)], [83Bi (1?)]} {[23Va (1+1)], [41Nb (1)], [73Ta (2+0)]} {[59Pr (1)]}
(6) Group VI:
{[8O (3+0)], [16S (4+0)] & [34Se (5+1)], [52Te (6+2)], [84Po (5+0)]} {[24Cr (4+0)], [42Mo (6+1)], [74W (4+1)]} {[60Nd (5+2)]}
(7) Group VII:
{[1H (2+0)], [9F (1)], [17Cl (2+0)] & [35Br (2+0)], [53I (1)]} {[25Mn (1)], [43Tc (0+0)], [75Re (1+1)]} {[61Pm (0+0)]}
(8) Groups 0+VIII:
{[2He (2+0)], [10Ne (3+0)], [18Ar (3+0)] & [36Kr (6+0)], [54Xe (8+1)]} {[26Fe (4+0)], [44Ru (7+0)], [76Os (6+1)]} {[62Sm (5+2)]}
61 (9) Group IX: {N, N, N , N, N} {[27Co (1)], [45Rh (1], [77Ir (2+0)]} {[63Eu (1+1)]}
(10) Group X: {N, N, N , N, N} {[28Ni (5+0)], [46Pd (6+0)], [78Pt (5+1)]} {[64Gd (6+1)]}
(11) Group XI: {N, N, N , N, N} {N , N, N} [65Tb (1]}
(12) Group XII: {N, N, N , N, N} {N , N, N} {[66Dy (7+0)]}
(13) Group XIII: {N, N, N , N, N} {N , N, N} {[67Ho (1)]}
(14) Group XIV: {N, N, N , N, N} {N , N, N} {[68Er (6+0)]} To proceed with new insights and analysis some explanations of Survey 2 are necessary as it follows. I. In front of the sign "+" is the number of stable, while behind the "+" is the number of unstable primordial isotopes. II. The order within the groups: first come intransitive elements (s or p),16 then the first transitive (d), and finally the second transitive element (f). With dark shadow tones are designated the stable elements; a total of 36: 13 odd in the odd groups and 23 even in the even groups (Survey 3b & 3c); with light tones are unstable elements17, a total of 25: the 8 odd in odd groups and the 17 even in even groups (Survey 3b on the left). Unshaded are the 20 stable "monoisotopic" elements (19 odd in odd groups, and 1 even in the even group, Be, in the second one); also unshaded are 3 radioactive elements (Tc, Pm, Po).
16 In front of the sign '&' are elements of the short periods, followed by elements of large periods. 17 S O P # ' # ' ve unstable primordial isotopes one or more.
62 III. The presented structure of PSE is characterized by the following regularities: 1. Odd groups contain odd, and even groups contain even elements; 2. Elements of odd groups have 2 or less stable isotopes; 3. Elements of even groups, in short periods have 4 or less, while in long periods have 4 or more stable isotopes; 4. Elements of short periods have only stable isotopes (maximum of 0 unstable primordial isotopes); 5. Elements of the odd groups of long periods have maximal 1 unstable primordial isotope; 6. Elements of even groups of long periods have maximal 2 unstable primordial isotopes.
2.4. Specific algebraic patterns Surveys 3a, 3b and 3c contain two key results of this research, first on the genetic code, and second one on the chemical code. Survey 3a contains the solutions of the system of two linear algebraic equations (shaded part of Survey), which appear to be in a full accordance with the distribution of codons in the genetic code through coding for 2, 4, 6 and 8 amino acids (Table 5): the 25 codons encode the amino acids of the less complexity (2AAs+4AAs) [(GP)+(ALVI)] which have only carbon and hydrogen (glycine M only hydrogen!) in the side chain; and 36 codons encode the AAs of greater complexity which have, except C and H, some other elements (N , O or S). The number of codons for encoding less complex AAs corresponds to the solutions of the first linear equation (x1 = 8 and y1 = 17): two nonstandard hydrocarbon AAs (GP) are encoded with 8, and four standard hydrocarbon AAs (ALVI) with 17 codons. On the other hand, the number of codons for encoding more complex AAs corresponds to the solutions of the second linear equation (x2 = 10 and y2 = 26): six AAs (CMFYWH) which do not have mapping of functional groups from the "head" to the "body" (side chain), are encoded with 10, and the eight AAs (STDENQKR), which from the "head" to the "body" are encoded with 26 codons (Survey 3a).
2.5. Key relationhips between GC and PSE Now we go to the PSE. The solutions of the system of two linear equations (in the shaded part of Survey 3b) are in an almost wholly accordance with the distribution of chemical elements (in terms of stability/instability and odd/even parity) into periods and groups. From a total of 61 multi-isotope elements, the 25, except stable, possess unstable primordial isotopes (light shaded tones in Survey 2); and 36 multi-isotope elements possess only stable elements (they do not have unstable primordial isotopes) (dark shaded tones in Survey 2).
63 Survey 3a Survey 4 2 y2 - x2 = 4 2 2 y1 - x1 = 3 y2 + x2 = 6 2 Z = 51 y1 + x1 = 5 x2 = 10 x1 = 8 y2 = 26 Y/4 = 51 y1 = 17 +
Survey 3b Survey 3c a b c d 101 10 91 y2 - x2 = 00 26 (16) y1 - x1 = 1 1 y2 + x2 = 36 01 11 25 (14) y1 + x1 = 25 02 12 24 (12) 03 13 23 (10) x2 = 13 x1 = 8 04 14 22 (08) y2 = 23 y1 = 17 05 15 21 (06) 06 16 20 (04)
The systems of linear equations as the scenarios for generating GC and PSE are given in Survey 3a and 3b, respectively; in Survey 3c there is an additional scenario for PSE with the showing why x2 = 13 and y2 = 23; that comes from the fact that only in the fourth row of Surveys 3c we have (03, 13, 23), a determination with the principles of similarity and self-similarity; columns: a = b - 10 and d = c - b; columns "b" and "c" correspond to x2 and y2' / A0/ 9 D ? 9 )64' > (Other details in the text).]
Further distributions are carried out through distinctions into odd and even elements M the odd elements within the odd groups and the even elements within the even groups, in both cases are in accordance with the model (the shaded part in Survey 3b). In accordance with the solutions of the first linear equations of (x1 = 8 and y1 = 17), the 8 unstable and odd elements are within the odd, and 17 unstable and even elements within the even groups. On the other hand, according to the solutions of the second linear equation (x2 = 13 and y2 = 23), the 13 stable and odd elements are in odd and 23 stable and even elements in even groups. [Under the notion "unstable" we mean the chemical element, which in addition to stable isotopes, have at least one unstable primordial isotope.]
64 2.6. Some additional observations on Genetic code At the end of the presentation of these researchMinsightsMresults, we once again return to the genetic code. Let us look at Table 3 and Table 4, in relation to Eq. (4). In Table 3 we have: monads M the numbers that appear at once; diads M twice, and triads M three times. At dark tones are the amino acids handled by enzymes of class I aminoacyl-tRNA synthetases and at light tones the amino acids handled by enzymes of class II aminoacyl- tRNA synthetases. [One can notice here a correspondence with PSE in Table 2 and with the mathematical expression in Eq. (2).]
The algorithm for "selection" of numbers in Tab. 4, going from bottom is as follows: in the first step were selected 2 numbers going up, and 0 numbers going sideways, that is 2 vs 0 (17 & 18 versus nothing & nothing); in the second step: 2 vs 2 (7 & 14 versus 13 & 15); third step: 2 vs 4 (1 & 4 versus 5 & 10 and 8 & 11); in the fourth step would be 2 vs 6, etc. (Eq.4).
In Table 4 we actually have the relationships within the atom number arrangement in Table 3, which arrangement corresponds with the mathematical expression in Eq. (2) in the next sense: 1 set of monads, 1 set of diads, 1 set of triads; at the same time: within the set of monads each number appears just once, within the set of diads M twice, and within the set of triads M three times. On the other hand, the "choices" presented in Eq. (4) are agreed with specific algorithm which corresponds with Generalized Golden Mean (GGM), through "metallic means family", for q = 2, 6, 12, 20, ... (Rako64' >>9' 97). With first choice there are 2 number-patterns: 18 & 17; with the second one there are 6 number-patterns: (18 & 17) + [(14 & 07) + (13 & 15)]; with the third choice there are 12 number-patterns: (18 & 17) + [(14 & 07) + (13 & 15)] + [(04 & 01) + (05 & 10) + (08 & 11)]. With this third choice there are all number-patterns for the number of atoms in amino acid molecules, within their side chains (12 patterns in Table 4 follow from 20 numbers, presented in Table 3). By all this, we can notice a regularity, valid for the GGM series, expressed in Eq. (4) and related to Eq. (5).
0 2 6 12 20 30 ... 2 4 6 8 10 .... (4)
[GP (27) + ALVI (76) + CMFYWH (128) + STDENQKR 153) = 204 + 180 2 4 6 8 .... (5)
65 Remark 7. Within four diversity types of 2-4-6-8 AAs there is the number of atoms as follows. Within the "heads" 180 atoms: I.GP 18; II.ALVI 36; III.CMFYWH 54; IV.STDENQKR 72); within the side chains 204 atoms: I.GP 9; II.ALVI 40; III.CMFYWH 74; IV.STDENQKR 81; in total, as in Eq. (5) where we have an example of self-similarity: in two inner groups, with 10 AAs, there are 204 atoms as in 20 side chains. On the other hand, in two outer groups, also with 10 amino acids, there are 180 atoms as in 20 amino acid heads. There is also a significant relation to the system, presented in Survey 4: [I.GACNP (26); II.SDTQH (42); III.YMEVL (59); IV.WRFIK (77)], which system follows from a unique arithmetical arrangement of natural numbers from 01 to 99 (Table 4 in relation to Figure 3 in Rako64' 2011b).
Remark 8. If the Multiplication table in the decimal numbering system is written in an adequate way (Table 1.1 in: Rako64' > ' p. 822) then the numbers per diagonal: 0, 2, 6, 12, 20, 30, ..., are also recognized as values for q in the equation of the Generalized golden mean x2 + px = q (q = 0, 2, 6, 12, 20, 30, ...) in the set of the family of "metallic means" (Spinadel, 1998, 444C )64' >>98 I natural numbers: (0, 1), (1, 2), (2, 3), (3, 4), (4, 5), ...; [First family of "metallic means": "golden mean" (p = 1; q = 1); "silver mean" (p = 2; q = 1); "bronze mean" (p = 3; q = 1), ... ; the second family of "metallic means": "golden mean" (p = 1; q = 1); "copper mean" (p = 1; q = 2), "nickel mean" (p = 1; q = 3, ...).]
Remark 9. Double values of the numbers found on the diagonal of Multiplication table: 0, 4, 12, 24, 40, 60, ..., are also recognized as values of one of the cathetus of Diophantus' triangles: 0. (1, 0, 1), 1. (5, 4, 3), 2. (13, 12, 5), 3. (25, 24, 7), 4. (41, 40, 9), 5. (61, 60, 11), ..., etc. Knowing above presented connections with the generalized golden mean, and that the genetic code is also determined by the golden mean (Rako64' 4458' # % 0 )' according to which the coding of four-codon amino acids "goes" over the Pythagorian triplet (5, 4, 3) (Shcherbak, 1994).
Remark 10. From the fact that the logic of the choices of the quantities of the number of # I TI 398' I 3>' ' 7' ' > G8 corresponds to the generalized golden mean and Diophantus' triangles at the same time, it follows that the choice of the quantitees of the number of atoms is also determined by both M by the generalized golden mean and by the set of Diophantus' triangles. [Quantities of the number of atoms: all different in monads; more quantities are the same in diads, as well as in triads, in Table 3.]
Remark 11. From the fact that Eq. (4) corresponds with Eq. (5) through the sequence of the first even numbers (2, 4, 6, 8), from the series of natural numbers, it follows that classification of amino acids into presented four classes is also determined by both M by the generalized golden mean and by the set of Diophantus' triangles.
66 Table 3. The number of atoms in the protein AAs (in their side chains)
Monads Diads Triads G 01 C 05 S 05 T 08 P 08 N 08 A 04 V 10 E 10 Q 11 M 11 H 11 D 07 L 13 I 13 F 14 Y 15 K 15 R 17 W 18
(Explanation in the text) Table 4. The relationships to the atom number arrangement in Table 3
1 2 3 01 05 08 3 04 10 11 3 07 13 2 14 15 2 17 1 18 1 6 4 2 (12)
6 8 6 (20)
(Explanation in the text)
67 Table 5. The standard genetic code with new design 2nd letter 1st 3rd U C A G UUU UCU UAU UGU C U F Y UUC UCC UAC UGC C U S uua UCA UAA UGA CT A L CT uug UCG UAG UGG W G cuu ccu CAU CGU U H cuc ccc CAC CGC C C L P R cua cca CAA CGA A Q cug ccg CAG CGG G auu ACU AAU AGU U N S auc I ACC AAC AGC C A T aua ACA AAA AGA A K R AUG M ACG AAG AGG G guu gcu GAU ggu U D guc gcc GAC ggc C G V A G gua gca GAA gga A E gug gcg GAG ggg G
The design responds to the classification of protein AAs into four classes, correspondently with four diversity types. The first diversity type (GP): the 8 codons in small non-bolding letters; second type (ALVI), the 17 codons in small bolding letters; third type (CMFYWH), the 10 codons in large letters and light shadow tones; fourth type (STDENQKR): the 26 codons in large letters and dark shadow tones. The three codons which are cross out, are the "stop" codons (cf. Survey 3a and Survey 4 in this chapter # )64' > = 9' TI ? Eq. 4 on pp. 826M827; Table. 6 on p. 829). 3. Discussion
For discussion of the obtained results, it is worth mentioning the current understanding of isotopy science, and hence we quote the comment of IUPAC, supplied with the official "Periodic Table of Isotopes":
"Standard atomic weights are the best estimates by IUPAC of atomic weights that are found in normal materials, which are terrestrial materials that are reasonably possible sources for elements and their compounds in commerce, industry, or science. They are determined using all stable isotopes and selected radioactive isotopes (having relatively long half-lives and characteristic terrestrial isotopic compositions). Isotopes are considered stable (non-radioactive) if evidence for radioactive decay has not been detected experimentally" (IUPAC Project 2007- 038-3-200, "Development of an isotopic periodic table for the educational community", October 1, 2013, www.ciaaw.org).18
18 "Dec 2016 M Project update published in Chem. Int. Nov 2016, p. 25; https://doi.org/10.1515/ci-2016- 0619 (Page last updated 11 Jan 2017)" 68 It is understood that the discussion of isotopes of chemical elements within the scientific community is strictly limited to planet Earth, because that is what we do know from the experiments. However, if it is possible to find a theoretical model, an arithmetic- algebraic scenario, that is, at a given moment of evolution of scientific knowledge, agreed with the distribution of chemical elements in the PSE, in terms odd-even parity of elements and their isotopes stability (moreover, agreed with corresponding distributions in the GC), then it should be that this model is valid for the whole universe, due to the fact that the content of the universe consists of one and the same type of chemical elements and their isotopes. However, having in mind the fact that the stability/instability of isotopes is experimentally determined (certainly by the best and most reliable laboratories, in terrestrial conditions), it must be said that the presented accordance with the models in Surveys 3a, 3b and 3c, although one hundred percent, or near to be one hundred percent, it is only one very distinctive trend, which should serve for further researches in both directions: for checking experimental results, and parallel with this, for checking the accordance with the models presented in Surveys 3a, 3b and 3c. In the current state of affairs, what may possibly spoil a 100-percent accordance, it might be in the next. In relevant literature one can see that potentially three of ten isotopes of tin are radioactive, but have not been observed to decay. One of these three actually has been identified as Sn-124, and in IUPAC sources is indicated its half-life decay larger than 110 17 years. In addition, one of the seven (stable) isotopes of Gadolinium (Gd-152) we take to be unstable with a half-life decay of 1.08 1014 years, while the isotope (Gd-160) with a half-life decay larger than 1.3 1021 remained in a stable status. Overall, we marked the isotopic state of Gadolinium (6+1). These are only two our interventions in relation to the IUPAC document which are, as follows from the above, also consistent with the understanding of isotopy in current science.19
3.1. Another understanding of the periodic system For an additional understanding of the PSE shown in Table 1 in the form of 6 periods and 14 groups, it is necessary to include a specific variant of the PSE which is inversely related in the sense that it consists of 14 periods and the 6 groups (Table 2). The arrangement of elements in the PSE in Table 2 is such that the odd elements are in odd and even elements in even groups, in the same way as in the PSE presented in
19 In IUPAC document M IUPAC Project 2007-038-3-200, "Development of an isotopic periodic table for the educational community" (October 1, 2013 www.ciaaw.org), for Europium was indicated the state (2 + 0), what means that both its isotopes are stable. However, in "New interactive, electronic version of the IUPAC Periodic Table of the Elements and Isotopes" it is modified so that now it is EU (1 + 1) because EU-151 is unstable and EU-153 the stable isotope, as it is in our system in Survey 2 [Eu-63 (1+1)].
69 Table 1. From this it follows that the same regularities, indicated in the explanations in Survey 2 (Section III), are valid for both, PSE in Table 1 as well as in Table 2. In addition, here we also find additional regularities. For them it is necessary to note that the arrangement of the PSE in Table 2 can be read in two ways: first, as indicated monads, diads and triads of columns, i.e. groups; and second, as odd and even groups (I, II, III, IV, V, VI). Correspondent comparisons (the left column of diads vs right column; left column of triads vs right; then, the first column vs third, the second vs fourth, etc.) show that the trend in differences, in the number of naturaly occurring isotopes, is such, that these differences, in 10 or more cases (from total 14), is less than 2. A maximum of two times, there are differences of 2 or 3 isotopes; altogether, all differences are within the frame of the set {0, 1, 2, 3}, that is of a "logical square".
3.2. Another similarities From Table 4 it is self-evident that here we have two solutions: the number of molecules-patterns in 20 canonical AAs is 6-4-2 as the sum of number-patterns in monads-diads-triads; on the other hand real sum of amino acid molecules is 6+8+6 = 20. `757O 757 / ' # the number of protons in 20 amino acids (within their side chains), what means a similarity and self-similarity. If we exclude the number of hydrogen atoms, that is hydrogen protons (58 + 59 = 117) of all 20 side chains, then remain 569 protons + 569 neutrons. At the same time, from Sukhodolets' system (Table C.2 in this chapter in relation to Table 7, p. 830 in )64' > ) follows a symmetrical division of AAs into two sub-classes with 58 and 59 hydrogen atoms, respectively.
3.3. Some additional observations The simplest chemical elements as nonmetal atoms, in the PSE, are: H + C + N + O ; 4 simplest nonmetal atoms ...... (6) Their simplest compounds are methane, ammonia, water and carbon monoxide:
CH4 + NH3, + H2O + CO ; 4 related simplest molecules ...... (7) Their functional groups are:
CH + NH2, + OH + CO ; 9 atoms ...... (8) The first possible organic molecule which takes into account all these groups is the simplest possible amino acid M glycine.
The' # ) 0 )R "mathematical trace" for nucleon number in 9 atoms:
CH 13 + NH2 16 + OH 17 + CO 28 = 74 (2 x 37) ...... (9) So much, in fact, there are nucleons in the "head" of amino acid (35 protons and 39 neutrons).
70 To better understand the above presented facts and analogies, in a future research it would be also the worth to take into account that both, the PSE and the GC, satisfy the golden mean property.20 [The GC satisfies the golden mean, over the set of all atoms 3 )64' 445' 2011b) as well as through the number of carbon atoms in 20 amino acids (Yang, 2000).] This should in particular be kept in mind for further testing "a new standpoint for addressing questions of selection vs random drift in the evolution of the code" (Swanson, 1984).
4. Conclusion
Considering the periodic system of chemical elements as a chemical code, then one can observe certain analogies between this code and the genetic code. This chapter is just devoted to analogies between these two codes. The presented researchMinsightsMresults show that the key to these analogies are specific relationships. Namely, within the genetic code there are exactly 61 codons wich encoding amino acids, plus 3 stop codons, plus 20 protein amino acids. On the other hand, within the chemical code there are exactly 61 entities in the form of stable isotopes, plus 3 unstable isotopes (Technetium, Promethium and Polonium), plus 20 `non-isotopeO entities (20 "monoisotope" elements). Thus, altogether there are 84 entities within the genetic as well as 84 entities within the chemical code. Moreover, the obtained analogies contain also a deeper connection between the genetic code and the periodic system of chemical elements. For example, inside 61 codons 25 of them encode the amino acids of less complexity (GP+ALVI), and 36 codons encode the AAs of greater complexity (CMFYWH+STDENQKR). Analogies can be further expanded by splitting less complex and more complex sets of AAs into subsets (indicated in parentheses). The distribution of codons is related to solutions of the system of two linear equations presented in the upper part of Survey 3a (8 codons for GP and 17 codons for ALVI equals 25 codons; 10 codons for CMFYWH and 26 codons for STDENQKR equals 36 codons) (Table 5). Analogies with chemical code are as follows. From total 61 multi-isotope elements, 25 of them, except stable, possess unstable primordial isotopes; in accordance with the solutions of the first linear equation (x1 = 8 and y1 = 17), in Survey 3a, where the 8 unstable and odd elements are within the odd, and 17 unstable and even elements within the even groups. On the other hand, according to the solutions of the second linear equation (x2 = 13 and y2 = 23), in Survey 3b, where the 13 stable and odd elements are in odd and 23 stable and even elements in even groups.
20 $ / ' = . )/ < ' 45 C )64' 1998a. About $ $E = )64' 445' > 9
71 In addition to the analogies of the number of isotopes, and the number of codons, there is an analogy more through the number of the chemical elements (in the arrangement of PSE with 6 groups - Table 2) and the number of atoms in the 20 protein AAs (Table 4): in both cases the order is made through the monads, diads and triads M with the singlets, doublets and triplets, respectively. At the end, there is a sense to assume that regularities in the genetic code are caused, at least partially, by regularities in the periodic system of chemical elements.
Appendix A. Two amino acid classes: four-codon and non-four-codon AAs
Table A.1. The Rumer's classification of amino acids into two classes
114 30 (119) 89 125 116 108 Gly GG (6) Phe UU (4) Leu Pro CC (6) Asn AA (4) Lys Arg CG (6) Ile AU (4) Met Ala GC (6) Tyr UA (4) ct
Thr AC (5) His CA (5) Gln Val GU (5) Cys UG (5) Trp Ser UC (5) Asp GA (5) Glu Leu CU (5) Ser AG (5) Arg
125 36 (120) 84 114 106 118 330-66 330±00
125 + 114 = 239 125 M 114 = 11
The four-codon amino acids are on the left and the non-four-codon amino acids on the right. Each of two classes is classified into two subclasses, corresponding to the number of hydrogen bonds in the nucleotide doublets. At the bottom (shaded) M the number of atoms in the amino acid molecules (side chains): within 32 amino acid molecules on the left and 29 on the right, within the set of "61" amino acid molecules, each molecule encoded by one codon. [There are the same results, 330-66 (as 8 x 33) and 330 (as 10 x 33) in standard GCT, both in relation with the result (9 x 33)±1 as we have shown in a previous work (Rako64' >>9' ? 98] Up/down: 119/120 atoms in 11/12 amino acids (side chains), respectively, within the set of "23" amino acids. [Calculations: (30+89 = 119); (36+84 = 120); (30+84 = 114); (36+89 = 125).] Up/down: (116/108 // 106 / 118) as the number of atoms within nucleotide doublets. (Note: In originsl Rumer's Table only the number of hidrogen bonds is calculated; all other calculations are ours.) 72 Table A.2. The Shcherbak's Table of multiples of "Prime quantum 037" (Table 1 in: Shcherbak, 1994).
The Shcherbak's basic Table indicates that the determination of the number of nucleons within amino acid molecules occurs by the multiples of number 037 (as a "Prime Quantum"), by the numbers with the same digits (111, 222, 333 etc.) or by the permutations of the obtain different cipher multiples.
Survey A.1. Relationships between AAs in Rumer's Table in accordance with the quantum-block "Aufbau" principle
mn an 1 2/ 1 AR = 21 T = 08 30 - 01 2 2/ 1+1 AR = 21 G + P = 09 30
3 2+1/ 1 VL + S = 28 T = 08 36 4 2+1/ 1+1 VL + S = 28 G + P = 09 36 + 01
5 2+1/ 2+2 IM + Y = 39 FL + NK = 50 89 6 2+1/ 2+2+2 IM + Y = 39 HQ + DE + SR = 61 89 + 11
7 2/ 2+2 CW = 23 FL + NK = 50 84 - 11 8 2/ 2+2+2 CW = 23 HQ + DE + SR = 61 84
[30 + (84 M 11) = 103] [(30 -1) + 84 = 113] [36 + (84 M 11) = 109] [30 + 89 = 119] [(30-1) + 84 = 116] (30 -1) + (89 + 11) = 129 [(36+1) + 89 = 126] Total atom number 239 [aln + 2(n-aln) = 236]
In the first four rows there are four-codon AAs (from the left side in Table A.1) and in the second four rows non-four-codon AAs (from the right side in Table A.1). In the first case, the "blocks" of AAs differ by 01 atom, and in the second case by 11. The designations: mn M number of molecules; an M number of atoms; at the bottom:
73 "aln" for AAs of alanine stereochemical type; "n-aln" for AAs of non-alanine / 3 )64 < *)4' 4478C "Total atom number" M the number of atoms within 23 AAs (side chains) in standard GCT.
Survey A.2. Relationships between Rumer's Table in Table A-1 and our Table in Survey 1
Quantum "30 - 01" Quantum "89+11"
(ac) T (dn, right) 6 (ca, ga, ag) HQ, DE, SR (dn, right) 129 1 (gc, cg) A, R (up, right) (au, ua) IM, Y (up, right)
Quantum "30" Quantum "89"
2 (gg, cc) G, P (up, left) 5 (uu, aa) FL, NK (up, left) 119 (gc, cg) A, R (up, right) (au, ua) IM, Y (up, right)
Quantum "36" Quantum "84" (ac) T (dn, right) (ca, ga, ag) HQ, DE, SR (dn, right) 3 8 120 (gu, cu, uc) V, L, S (dn, left) (ug) CW (dn, left)
Quantum "36 + 01" Quantum "84 -11" 4 7 (gg, cc) G, P (up, left) (uu, aa) FL, NK (up, left) 110 (gu, cu, uc) V, L, S (dn, left) (ug) CW (dn, left)
The whole arrangement is in relation to the arrangement of AAs in Survey A.1. On the left are four-codon AAs and on the right non-four-codon AAs (as in Table A.1). The designations "up", "dn" (down), "left" and "right" refer to the positions in Survey 1. Amino acids at the positions indicated by the numbers 1-8 correspond to the amino acids at the same positions in Survey A-1.
74 Survey A.3. Relationships between the rows in Survey A.1
(1M6) (30 M 1) + (89 + 11) = 129
(1M5) (30 M 1) + (89 + 00) = 118
(2M6) (30 M 0) + (89 + 11) = 130 129 (2M5) (30 M 0) + (89 + 00) = 119 119 109 130 (3M8) (36 + 0) + (84 + 00) = 120 120 110 (3M7) (36 + 0) + (84 M 11) = 109
(4M8) (36 + 1) + (84 + 00) = 121
(4M7) (36 + 1) + (84 M 11) = 110
The relationships are such that they show changes for the unit in the first, second and / or third position in the record of the number of atoms in the amino acids (in their side chains). At the same time, it is obvious that some sequences are generated from the series of natural numbers (two last columns).
75 Appendix B. Two amino acid classes: AAs handled by class I and AAs handled by class II of enzymes aminoacyl-tRNA synthetases, respectively
Table B.1. E $ E 3 )64' 2007a)
This new Table of the genetic code is realized in a specific combination of standard GCT and Swanson's Codon Path Cube (Swanson, 1984). In the combination are three different permutations from the total of 24. Bold positions (and dark tones): codons coding for AAs handled by class II of enzymes aminoacyl-tRNA synthetases (aaRS); non-bold positions: codons coding for AAs handled by class I of aaRS plus three "stop" codons, denoted with asterisk. As it is obvious, two groups of AAs are completely separated [Class I: (LVIM)+(YQERWC); Class II: (FSPTA)+(HDNKG)]. For the process of combining, the agreement-disagreement principle (Section 1.2) is valid as follows. In the standard GCT for all three positions in the codon (first, second and third letter), we have the maximal agreement M the same permutation (UCAG) all three times, but there is no
76 agreement in the distinction into two classes in two separated areas (Wetzel, 1995). In the Codon Path cube the agreement in permutations is lower: for the first and second positions the UCGA and for the third position CUAG permutation; lower agreement in permutations, but greater in terms of separation into two classes of AAs, almost complete separation with one exception 3 )64, 1997a). Finally, if we combine the GCT and Codon Path Cube, in the way shown here, we have a more lower agreement on permutations, but the greater agreement regarding the separation into two classes, they are completely separated. Apparently, this is also an analogy with the Heisenberg principle of uncertainty. [UCAG is used in the inner part of the Table for the first letter, and in the external part is UCGA permutation; for the third letter: in the inner part of the Table is UCGA, and in the external part CUAG permutation; UCGA permutation for both parts of the Table is valid for the second letter.]
Appendix C. Two amino acid classes: AAs with odd and AAs with even number of hydrogen atoms
Table C.1. The Sukhodolets' Table of the number of hydrogen atoms
The number of hydrogen atoms (n) within amino acid molecules, in relation to natural numbers series: 5, (6), 7, 8, 9, 10, 11, 12, 13, 14 (Sukhodolets, 1985). First letter plus second letter equals a codon root (nucleotide doublet). The codon root plus third letter equals a complet codon. [Cf. this regularity of the hydrogen atoms within amino acids (in relation to natural number sequence) with the regularity of total number of atoms within nucleotides in standard GCT: within 64 codons (192 nucleotides) there are 2 x 3456 atoms; 3456 in two inner as well as in two outer columns.]
77 Table C.2. The Sukhodolets' Table of the number of hydrogen atoms ( )64' > 8
The number of H atoms (in brackets) and nucleons G (01) 01 A (03) 15 S (03) 31 D (03) 59 C (03) 47 (13) 153 N (04) 58 P (05) 41 T (05) 45 E (05) 73 H (05) 81 (24) 298 (59/58) Q (06) 72 V (07) 43 F (07) 91 M (07) 75 Y (07)107 (34) 388 569/686 W (08)130 R (10) 100 K (10) 72 I (09) 57 L (09) 57 (46) 416 569 as neutron number and 686 as proton number! 569 M 59 = 627 M 117 686 M 58 = 628 Table C.2. The Sukhodolets' system of amino acids in a 4 x 5 arrangement. The Sukhodolets' Table, with a minimal modification (Sukhodolets, 1985): the system of 4 x 5 AAs. The shadow space: AAs with even number of hydrogen atoms (4, 6, 8, 10); the non-shadow space: AAs with odd number of hydrogen atoms (1, 3, 7, 9, 11). In brackets: number of hydrogen atoms (within amino acid side chain) and out of brackets the number of nucleons. Nucleon number through a specific "simulation": 569 nucleons within two outer rows, as the number of neutrons, 569, in all 20 AAs M within their side chains; and 686 nucleons within two inner rows, as the number of protons, 686, in all 20 AAs M within their side chains. Within 20 side chains of amino acid molecules there are 569 neutrons as well as 569 non-hydrogen protons. Within 20 side chains of amino acid molecules there are 117 hydrogen protons, what means 117 hydrogen atoms at the same time (117 = 59 + 58).
Appendix D. O' ((!)! *P (IUPAC document)
OH& += > 6-039-2-200; Start Date: 18 December 2015; Division Name: Inorganic Chemistry Division; Division No.: 200
Objective This project will deliver a recommendation in favor of the composition of group 3 of the periodic table as consisting either of
1. the elements Sc, Y, Lu and Lr, or 2. the elements Sc, Y, La and Ac.
The task group does not intend to recommend the use of a 32-column periodic table or an 18-column. This choice which is a matter of convention, rather than a scientific one, should be left to individual authors and educators. The task group will only concern itselve with the constitution of group 3. Once this is established, one is free to represent the periodic table in an 18 or 32 column format. 78 Description
The question of precisely which elements should be placed in group 3 of the periodic table has been debated from time to time with apparently no resolution up to this point. This question has also received a recent impetus from several science news articles following an article in Nature magazine in which the measurement of the ionization energy of the element lawrencium was reported for the first time.
We believe that this question is of considerable importance for chemists, physicists as well as students of the subject. Students and instructors are typically puzzled by the fact that published periodic tables show variation in the way that group 3 of the periodic table is displayed. The aim of the project is to assemble a task group to make a recommendation to IUPAC regarding the membership of group 3 of the periodic table.
Various forms of evidence have been put forward in support of each version of group 3. In the basis of this evidence arguments have been proposed while appealing to chemical as well as physical properties, spectral characteristics of the elements and criteria concerning the electronic configurations of their atoms. The task force will aim to evaluate all this evidence in order to reach a conclusion that encompasses these different approaches. Progress March 2016 M Project announcement published in Chem. Int. March 2016, p. 22; http://dx.doi.org/10.1515/ci-2016-0213 H 4 - > 7P (Original text from IUPAC document)
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82 Chapter 3
Chemical Distinctions of Protein Amino Acids within Genetic code
In this chapter it is shown that 20 protein amino acids ("the canonical amino acids" within the genetic code) appear to be a whole and very symmetrical system, in many ways, all based on strict chemical distinctions from the aspect of their similarity, complexity, stereochemical and diversity types. By this, all distinctions are accompanied by specific arithmetical and algebraic regularities, including the existence of amino acid ordinal numbers from 1 to 20. The classification of amino acids into two decades (1-10 and 11-20) appears to be in a strict correspondence with the atom number balances. From the presented "ideal" and "intelligent" structures and arrangements follow the conclusions that the genetic code was complete even in prebiotic conditions (as a set of 20 canonical amino acids and the set of 2+2 pyrimidine / purine canonical bases, respectively); and the notion "evolution" of the genetic code can only mean the degree of freedom of standard genetic code, i.e. the possible exceptions and deviations from the standard genetic code.
From a chemical point of view the first step of classification of protein amino acids (AAs), "the canonical amino acids" within genetic code, must be the classification into aliphatic and aromatic AAs, where on a hierarchical scale of changes by similarity and complexity, aliphatic AAs must precede the aromatic. For the same reason of the chemical hierarchy, within the class of aliphatic AAs at the beginning must be the hydrocarbon AAs (possesing in the side chain carbon and hydrogen, or hydrogen only, in the case of glycine), and at the end two sulfur AAs, quite different from preceded non-sulphuric AAs. This means that two sulfur AAs [as last in class of aliphatic AAs] must be found in a direct contact to the aromatic ones. In further course of sequencing of AAs, in terms of changes by similarity, from the aspect of the AAs singlets and/or doublets, i.e. pairs, it should be considered appropriate distinctions in three areas: in the hydrocarbon, aromatic, and that between them. In the set of aromatic AAs, Phe came first, as the simplest, followed by Tyr, and Trp, all three with possession of a benzene ring. At the very end ultimately must be His, the only one who does not possess the aromatic benzene ring. In the set of hydrocarbon AAs, at the very beginning must be Gly as the simplest one. Follows Ala as the first possible case of hydrocarbon series with an open carbon chain. Then come Val and Pro, both with three carbon atoms in the side chain, rather than Leu and Ile with four carbon atoms. By this, Val with half-cyclic chain precedes Pro with cyclic one; also Ile precedes Leu, as more similar with Pro. [The details of the relationship between Val and Pro, see = )64 < *)4' 447C # H J' = )64 445' Survey 4, p. 290, where the pair Pro-Ile is one of ten pairs of amino acids classified into two classes: class I (with Ile), handled by class I of enzymes aminoacyl-tRNA synthetases, and class II (with Pro), handled by class II of enzymes aminoacyl-tRNA synthetases.]
83 Finally, it remains to determine the chemical distinctions of AAs in "between" area. We have already said that sulfur amino acid pair, Cys-Met, precedes aromatic amino acids. As chalcogene AAs, they must be in contact with other two chalcogene amino acids, Ser-Thr. By this, the contact have to be made via Cys because it possesses SH group, correspondent to OH group in Ser as well as in Thr. It is to be understood that a pair of oxygen AAs with the hydroxyl (OH) functional group in side chain must be in contact with a pair of two also oxygen AAs, but which possess the carboxyl (COOH) functional group: Asp-Glu. But the problem is that both of these AAs have their amide derivatives (Asn-Gln) and it is not easy, when determining the distinctions, determine which here precede and which ones follow. It turns out, however, that the problem easier to solve when returning to the beginning, in the area of hydrocarbon AAs, to the "point" of the pair Ile-Leu. Further must follow the pair of nitrogen derivatives, Lys-Arg, and Lysine must come first with four carbon atoms in the side chain, which number is also valid for Leucine; and then, with the validity of both principles M the continuity and minimum of change M comes Arginine with three atoms (not counting carbon atom in the guanidino group). Then, chemically speaking, it is very natural that after Arginine comes Gln with its precursor, the glutamic amino acid, both (Gln-Glu) with two carbon atoms in the side chain; it is naturaly indeed that, in terms of chemical similarity, after 3C atoms occurs changes into 2C atoms, better than into 1C atom, as the case we have in the pair Asn-Asp. [As in the case of the guanidino functional group in arginine, no carbon atom is counting in the carboxylic or amide functional group.] With this, chemical sequencing of series of 20 AAs closes, starting from the first, glycine, and ending with very different histidine. * The main result of this pure chemical sequencing of AAs, presented in Table 1, shows that these chemical distinctions are accompanied by specific arithmetical regularities, including the existence of amino acid ordinal numbers from 1 to 20, with two decades (1-10 and 11-20) ; and also shows the full balance of the number of atoms in the 20 amino acid molecules: 102 ± 0 atoms in two decades, as well as on two zig-zag lines, where such a system with two zig-zag lines represents the first possible periodic system with two periods. However, the result of the most surprising, is the result shown in Table 2. If we take the four by four AAs from Table 1 and line up they in five rows, we obtain an arrangement of AAs, as it is shown in Table 2 where the difference ratio in the number of atoms, per rows equals 11:11:11 [(61-50 = 11); (42-31 = 11), (31-20 = 11)]. But the surprise comes only with an insight into Table 3 in which, exactly in the 11th step of a specific number arrangement where we find the same arithmetical result (20-31-42-61-50) valid for the number of atoms in the arrangement of AAs in Table 2. [Notice that this sequence is in contact with two sequences which contain the
84 first pair of friendly numbers, 220-284, what is a coincidence only, or what more?]1 However, knowing a previous result 3 )64' > '8 where the number of atoms in four diversity types of the protein AAs is identical to the unique arithmetical result in a unique arithmetical arrangement (Remark 1), surprises for the presented atom number sequence no longer; simply, everything is brought into a conection with the three endpoints of this arrangement: 00-11-22, in the form of a mirror mapping (00-11-22 / 22-11-00) with compression in the mirror plane (00- 11-22-11-00).
Table 1. The chemically determined order of protein amino acids
(1) G 01 08 N (11) (2) A 04 07 D (12) (3) V 10 05 S (13) (4) P 08 08 T (14) (5) I 13 05 C (15) (6) L 13 11 M (16) (7) K 15 14 F (17) (8) R 17 15 Y (18) (9) Q 11 18 W (19) (10) E 10 11 H (20) 102 102 511 o/e 511
The 20 protein AAs, arranged into two decades in accordance to ordinal amino acid number, 1- 10 and 11-20; the numbers presented outer: the ordinal numbers 1-20; the numbers presented inner: the number of atoms within side chain of the responding amino acid. The designations OF" as "odd / even". For details see the text.
1 A hypothesis on the determination of the genetic code with the perfect and friendly numbers we have presented in the book ( )64' 44:8 (www.rakocevcode.rs) [Perfect numbers: 6, 28, 496, 8128, etc; the pairs of the friendly numbers: (220-284), (1184-1210), (17296-18416) etc.] 85 Table 2. The order of five quartets of protein amino acids
G (01) A (04) N (08) D (07) 20 V (10) P (08) S (05) T (08) 31 I (13) L (13) C (05) M (11) 42 K (15) R (17) F (14) Y (15) 61 Q (11) E (10) W (18) H (11) 50
51-1 51+1 51-1 51+1 The explanation in the text.
Table 3. A specific natural numbers arrangement
00 02 04 06 08 10 12 11 13 15 17 19 21 23 22 24 26 28 30 32 34 11 16 21 26 31 36 41 00 05 10 15 20 25 30 44 60 76 92 108 124 140 12 14 16 18 20 22 23 25 27 29 31 33 34 36 38 40 42 44 41 46 51 56 61 66 30 35 40 45 50 55 140 156 172 188 204 220 22 24 26 28 30 32 33 35 37 39 41 43 44 46 48 50 52 54 66 71 76 81 86 91 55 60 65 70 75 80 220 236 252 268 284 300 32 34 36 38 40 42 43 45 47 49 51 53 54 56 58 60 62 64 91 96 10 106 111 116 1 80 85 90 95 100 105 300 316 332 348 364 380 ...
The arrangement represents the Table of distinct 2-5 adding (TDA) with starting column which follows from TMA, explaned in Remark 1. All other explanations in the text.
86 Table 4. A specific protein amino acids arrangement
119 G 01 N 08 L 13 M 11 (33) A 04 D 07 K 15 F 14 (40) 120 V 10 S 05 R 17 Y 15 (47) P 08 T 08 Q 11 W 18 (45) I 13 C 05 E 10 H 11 (39) 117 G 01 N 08 L 13 M 11 (33) 24/13 18/23 40/39 37/43 118/119 (37) (41) (79) (80) 117/120 118
The first row is repeated at the bottom, and thus one cyclic system is obtained. There are 117 atoms in two outer columns; at even positions 118, at odd 119; in two inner columns 120 atoms. On the other hand, in the lower half of the Table there are 117 atoms ones more; in the lower diagonally "wrapped" area 118, and in the upper 119; in the upper half of Table 120 atoms. The repeated four AAs at the bottom of the Table make to achieve a diagonal balance with a difference of only one atom; moreover, to establish a sequence from the series of natural numbers: 117, 118, 119, 120. (About generating the Table see in the text.)
87 Table 5. The Rumer's classification of amino acids into two classes (Rumer, 1966)
114 30 (119) 89 125 116 108 Gly GG (6) Phe UU (4) Leu Pro CC (6) Asn AA (4) Lys Arg CG (6) Ile AU (4) Met Ala GC (6) Tyr UA (4) ct
Thr AC (5) His CA (5) Gln Val GU (5) Cys UG (5) Trp Ser UC (5) Asp GA (5) Glu Leu CU (5) Ser AG (5) Arg
125 36 (120) 84 114 106 118 330-66 330±00
125 + 114 = 239 125 M 114 = 11
The four-codon amino acids are on the left and the non-four-codon amino acids on the right. Each of two classes is classified into two subclasses, corresponding to the number of hydrogen bonds in the nucleotide doublets. On the other hand, nucleon number within amino acid molecules in these two classes is determined by Pythagorian triangle (Figure 1 in Shcherbak, 1994). At the bottom (shaded) M the number of atoms in the amino acid molecules (side chains): within 32 amino acid molecules on the left and 29 on the right, within the set of "61" amino acid molecules, each molecule encoded by one codon. [There are the same results, 330-66 (as 8 x 33) and 330 (as 10 x 33) in standard GCT, both in relation with the result (9 x 33)±1 as we have shown in a previous work (Rako64' >>9' ? 98] Up/down: 119/120 atoms in 11/12 amino acids (side chains), respectively, within the set of "23" amino acids. [Calculations: (30+89 = 119); (36+84 = 120); (30+84 = 114); (36+89 = 125); (125 M 114 = 11). ]; [(05 x 6 = 30±0); (15 x 6 = 89+1); (06 x 6 = 36±0); (14 x 6 = 84±0)]; Up/down: (116/108 // 106 / 118) as the number of atoms within nucleotide doublets. (Note: In originsl Rumer's Table only the number of hydrogen bonds is calculated; all other calculations are ours.)
Remark 1. There is a unique arrangement of the natural numbers series, such as in the "Table of minimal adding" (TMA), with changes in the values for 01 horizontally and just for 11 vertically [01, 02, ..., 09, 10, 11 / 12, 13, ..., 20, 21, 22 / 23, ' 44B 3 )64' > ' 9' p. 826). Moreover, in addition to horizontal and vertical arrangement, there is also a diagonal arrangement, which we point out here in particular (Box 1 in relation to Box 2). By this one can notice that the TMA correspons with the Periodic table of chemical elements (PT), through the validity of both Mendeleev principles: the principle of continuity and the principle of minimum 88 change. Since both these principles are also apply to the amino acid code (Swanson, 1984), all together support the hypothesis of a prebiotic complete amino acid code, i.e. genetic code 3 )64' >>98
The presented sequence also points to the logic of choice the uniqueness in the case of the selection of protein amino acids within the standard genetic code, minimal in two manner. So, in this chapter, from the sequence (00-11-22-11-00), located in the starting column of Table 3, generated from the initial triplet of the last column (00-11-22) in the "Table of minimal adding" (TMA), explained in Remark 1, follows: [00 + (2 x 10) = 20], [11 + (2 x 10) = 31], [22 + (2 x 10) = 42], [11 + (5 x 10) = 61], [00 + (5 x 10) = 50]. So, we get the sequence (20-31-42-61-50), analogical to the the number of atoms in five rows of 20 AAs (within their side chains) as in Table 2; There is also an other logic of choice the uniqueness, presented in a previous paper 3 )64' > 8 as follows: [(25-36-26-16) (16-17- 58B A37-5 = 01) (36-25 = 11)] [(26), (26+16 = 42), (42+17 = 59), (59+18 = 77)], we get the sequence (26-42-59-77), analogical to the number of atoms in four diversity types of 20 AAs (within their side chains), dispersed in four columns of a specific arrangement (Figure 3 in )64' > ). In addition to the above, there are other relationships with the amino acid (genetic) code. Within TMA there are two horizontally adjacent numbers, 5 and 6 (6 - 5 = 01) whose squares (25 and 36, respectively) are vertical neighbors (36-25 = 11); and this is a unique case, the only one in the whole TMA. Analogy (and correspondance) with the amino acid (genetic) code is in the sense that within the Table of genetic code (TGC) 25 codons are coded for (2+4) amino acids of lower complexity (possess C & H, or only H, in side chain), while 36 codons are coded for (6+8) amino acids of more complexity, which besides C & H, possess even more the atoms of other elements: N, O or S (cf. Table 6 in )64, 2011b, p. 829). On the other hand, in the sequence [(25-36-26-16) (16-17-18)] the sum of the three numbers with the status "to be added" is 16 + 17 + 18 = 51, which is a fourth of the sum of the four numbers in the given sequence: 26 + 42 + 59 + 77 = 204 = 4 x 51. And this is the only such case in the TMA. For example, such a regularity does not apply to the TMA analogue sequence [(36-47-37-27) (27-28-29)], because 27+28+29 = 84 is not a fourth of the sum of the four numbers in an analogously obtained sequence: 37 + 64 + 92 + 121 = 314 + 4 x 84. The "enigmatic" question why, in the arrangement of Table 3 (in stating column), the first three rows increase per 2 units, and the other two rows per 5 units, it stops to be mysterious when we know that the ratio of numbers 2 and 5, in binary form (010/101), represents the first possible connection between the "golden route" on Farey tree and the path of the permanent change 3> > > > > 8 / 3 )64' 4458 [The answer to the question why 3 rows ("per 2 units") versus 2 rows ("per 5 units") is currently not known. However, it is possible suppose that this could be due to the fact that "the limit of the golden numbers is 3/2" (Moore, 1994).]
89 Box 1. The logic of a parallel choice The important feature of the "Table of Minimal Adding" (TMA) (Figure 1), within the limits of the two-digit arrangement of decimal number system, is a specific diagonal structure, with 10 odd and 9 even of diagonals. Here, in Box 2, only odd diagonals are presented, because among them there is a diagonal (the last row in the second quadrant of Box 2), which analogically corresponds with a specific chemical distinct arrangement of protein amino 3D 4 )64' > ' 5?9C = I (U 8= 5Z5 AAs of alanine stereochemical type (within their side chains) there are 91 and 81 atoms, respectively; in 4+4 AAs of alanine stereochemical type plus 1+1 AAs of valine stereochemical type there are 71 and 61 atoms, respectively; in 4+4 AAs of alanine stereochemical type plus 1AA of glycine stereochemical type, plus 1 AA of proline stereochemical type there are 41 and 31 atoms, respectively. Finally, within the set of four AAs of non-alanine type, 2+2 AAs possess 21 and 11 atoms, respectively. [About four a / 3H' 4548 3 )64 < *)4' 447.)
Remark 2. The determination of the amino acid (genetic) code with the golden mean leads to the CIPS (Cyclic Invariant Periodic System), in which the positions of five classes of AAs are strictly determined, two in the less complex and three in the more complex superclass: 1. (SC- TM), 2. (GV-PI), 3. (DE-NQ), 4. (AL-KR), 5. (FY-HW). Less complex aliphatic AAs in the side chains, besides C & H (Gly only H), contain the less polar nitrogen atom (N), and no longer contain the polar atom of oxygen (O), while more complex AAs contain it. Aromatic AAs also fall into the superclass of more complex AAs. In addition, we recall that in all the presented AAs pairs, the first member, as a smaller molecule, belongs to class II of AAs, handled by class II of enzymes aminoacyl-tRNA synthetases, and the second one to class I. [Cf. Figs 6 & 7 in 3 )64' > ' 5?-833) and 4th quadrant within Box 2 in this chapter.]
90 ... (-2) ...... -22 (-1) -21 -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 (0) -10 -09 -08 -07 -06 -05 -04 -03 -02 -01 00 (1) 01 02 03 04 05 06 07 08 09 10 11 (2) 12 13 14 15 16 17 18 19 20 21 22 (3) 23 24 25 26 27 28 29 30 31 32 33 (4) 34 35 36 37 38 39 40 41 42 43 44 (5) 45 46 47 48 49 50 51 52 53 54 55 (6) 56 57 58 59 60 61 62 63 64 65 66 (7) 67 68 69 70 71 72 73 74 75 76 77 (8) 78 79 80 81 82 83 84 85 86 87 88 (9) 89 90 91 92 93 94 95 96 97 98 99 (A) A0 A1 A2 A3 A4 A5 A6 A7 A8 A9 AA (B) B1 B2 B3 B4 B5 B6 B7 B8 B9 BA BB
Figure 1. The Table of minimal adding in decimal numbering system in relation to Box 1 and Box 2. A specific arrangement of natural numbers in decimal numbering system, going from 01 to 11 and so on. [Notice that this arrangement is only a variant of the arrangement within Table 4 in Supplement 1.]
At the end of this analysis of the relationship between the three arrangements, presented in the three tables (Tables 1-3), we present another more, perhaps slightly hidden, relationship. Namely, from the relationship between the first two and the last two rows of numbers in the two upper structures within Box 2, it follows that there is a correspondence between the Table of Minimal Adding (TMA) and the Multiplication Table (TM), as well as between their diagonal structures in the decimal number system. On the other hand, if TM is written adequately, systemic and systematically,2 as TMA, then, on both sides of the main diagonal, there are the = >' ' 7' ' >' 3 )64' > 8' # @I@ I ; 30 ' 445' 444C )64' >>9' 2011b). It follows from this that it can be understood why the amino acid (genetic) code is / 3 )64' 445' > 8 3 ) 8
2 "Systemic and systematically", that is, correspondent with the principles of continuity and minimum change (which, as we said, apply to both M genetic code and chemical code; cf. footnote 3). Practically, this means writing first the sequence (0, 1, 2, ..., 9) as a vertical column, with the status of the first column. Then the results of the multiplication are written: the results of the multiplication by zero yield the zeroth column, to the left of the first column; the results of the multiplication by the unit give this same, start-up, t.e. the first column. Next, continuous columns of multiplication with 2, 3, etc
91 However, if we note that the said numbers on the main diagonal of TM, taken in a double amount (0, 4, 12, 24, 40, ...), represent the first cathete in the PythagoreanMDiophantine triangle set, and increased by one M the hypotenuse of the same triangles: 1, 5, 13, 25, 41, ... [The second cathete is "taken" from the set of odd natural numbers, respectively.] Thus, it is possible to understand why the amino acid (genetic) code was determined by the first (3-4-5) PythagoreanM Diophantine triangle (Shcherbak, 1994). In addition, considering the fact that there is a "coherence of the chemical and genetic code" 3 )64' 44 83, more precisely, there is an analogous unity of the chemical and genetic code ( )64, 2017), that is, if everything here presented, taken together, then there can be enough understanding for the standpoint that "the genetic code is an intelligent design ..." (Castro-Chavez, 2010, p. 718).4
Box 2. The arrangement followed from diagonal structure of TMA
(01) [1] [1] (99) (03,13,23) [3] [3] (77, 87, 97) (05, 15, 25, 35, 45) [5] [5] (55, 65, 75, 85, 95) (07, 17, 27, 37, 47, 57, 67) [7] [7] (33, 43, 53, 63, 73, 83, 93) (09, 19, 29, 39, 49, 59, 69, 79, 89) [9] [9] (11, 21, 31, 41, 51, 61, 71, 81, 91)
S05 T08 L13 A04 G01 31 M 41 11+ 91 = 102 (91 11 = 8 x 10) D07 E10 M11 C05 P08 21+ 81 = 102 (81 M 21 = 6 x 10) V10 61 31+ 71 = 102 (71 M 31 = 4 x 10) K15 R17 Q11 N08 71 41+ 61 = 102 (61 M 41 = 2 x 10) F14 Y15 W18 H11 I13 51 / 51 (The top two quadrants possess five diagonals 91 81 G V 11 each in the first and second part of the TMA, respectively. All other explanations in Box 1) P I 21
We again look at Box 2 (quadrant fourth), from the aspect of the viewing of possible distinct chemical pairs of AAs. We see that the pairs strictly correspond to the pairs found in two classes
3 @G e chemical code, built on the very principles mentioned and in complete accordance with the genetic code. G # / and cyclicity of the / G@ 3 )64' 44 ' 8 (Cf. footnote 10.) 4 "With insight into the results ... one is forced to propose a hypothesis (for further researches) that here, there really is a kind of intelligent design; not the original intelligent design, dealing with the question M intelligent design or evolution, which is rightly criticized .... Here, there could be such an intelligent design, which we could call R0 J " R 0HJ"8 # at design which was presented by F. Castro-Chavez (2010), and is also in accordance with the Darwinism" 3 )64' > 6, footnote 47) [Additional note: The evidence that Darwin tacitly advocates the idea of intelligent design, learns from the insight that his only illustration (Diagram) in Origin of Species is actually a strict mathematical program corresponding, directly and / or indirectly, to the structures currently known within Multiplication Table (TM) and within Table of Minimal Adding (TMA) [About DarwinR Diagram as a mathematical program one can see in (Rako64' 449C > 68B
92 of AAs handled by two classes of enzymes aminoacyl-tRNA synthetases (Wetzel, 1995; Rako64' 44:8 M -alanine type set, and in addition, in the quantum "11" and the quantum "21"; the result of matching is obvious: a larger molecule in class I, while lower in class II, GV and PI, respectively. Then we look at the pairs in quantums "81" and "91". In all cases, except in one, in one column there are larger molecules (class I), and in the second one smaller molecule (class II). [The exception is a par CM in quantum "81" where both AAs belong to class I, and a pair of ST in quantum "91" where both AAs belong to that class II (c 0/ 9 )64' 4458]5 In reactions catalysed by the class I aminoacyl-tRNA synthetases, the aminoacyl group is coupled to the 2'-hydroxyl of the tRNA, while, in class II reactions, the 3'-hydroxyl site is preferred, and these two positions differ in their polarity.6 This means that the ordered arithmetical quantities, described above, correspond to the arrangement of AAs from the aspect of fine nuances in the chemical polarity of molecular classes. However, in the case that the reader doubts that this is a matter of correspondance of the arithmetical and chemical-biological entities, we also note the fact that, in the natural selection of amino acid molecules, it was necessary to "skip" the quantity 51, and after the quantity 41 choose a quantity of 61; it further means that it was not possible to choose a lower quantity of a set of molecules, for example a small molecule of ornithine precedes a large molecule of arginine (in the genetic code) as Tomas H. Jukes assumed. (Jukes, 1973, p. 24= `J / O.) [Amicus Plato, sed magis amica veritas.] * Not only in two decades (as in Table 1), the balance of the number of atoms is contained also within the halves of both decades, in four quintets: 1. GAVPI, 2. LKRQE, 3. NDSTC, 4. MFYWH. However, the said regularities are expressed (become visible) only when the four quintets develop into a system of five quartets; the five upper and five lower pairs are arranged according to the next logic: the first pair of the upper quintet with the first pair of the lower one, the second with the second, etc. (Table 4). If in fact the resulting system (with five quartets) perceived as a cyclic system (the first row in a cyclization occurs also as the last), then the balances (and symmetry) exhibit a specific self-similar arrangement more: two times appear the same quantities of the number of atoms, although with the different qualities (with different AAs) in four cases (117, 117) (118, 118), (119, 119), (120, 120). Such a repetition means per se a correspondence with the principle of self-similarity (Remark 3).
5 These pairs are exceptions also in the basic chemical characteristic. The CM is the only pair in which both members, apart from atoms of the first and second periods of PT of chemical elements, possess one atom from the third period. On the other hand, the ST and CM pairs are chemically similar and constitute a set of halcogen AAs, over oxygen and sulfur, both from the sixth group of halcogen elements. With this insight, it is easier to understand why they are at the center of CIPS: they start counting from them and have an ordinal number 1. 6 Doolittle (1985, p. 76) has shown that a possiblereplacement of amino acids during the evolution of proteins must O ; R # ; P
93 Remark 3. So far, the chemical meaning for the duplicates of AAs in Table of Genetic Code (TGC) has been shown #= 0. 3E)' 4758C 0.J 3 )64' >>:' :C Wohlin, 2015, Table 2); GP-%J 3 )64 *)4' 447' 0/ 8C chapter: GNLM in Table 4. All these duplications, ipso facto, support the hypothesis of a prebiotic comp 3 )64' >>98' # # )# ("Canonical genetic code"), and all other genetic codes are "deviant codes" (Weaver, 2012, pp. 568-569), or exceptions from the standard genetic code. These codes, per se, do not change nothing on the fact, when it comes to the genetic code as "amino acid code" (Swanson, 1984), because they are represented in all variants with the same 20 protein amino acids. (The only exceptions are Pyrrolyisine and Selenocysteine, presented at some very few organisms.) [Knight ' >> ' 94= @ # D ' QR emerged before the last universal ancestor; subsequently, this code diverged in numerous nuclear and organelle lineages".]
But, not only that, the pair 119-120 stand in relation to the same pair in the Rumer's Table of AAs and their corresponding nucleotide doublets (Table 5); The number of 118 atoms is also found as a half number of atoms within a "specific amino acid arrangement" ( )64 < *)4' 1996, Survey 1);7 and 117 atoms as the number of hydrogen atoms in all 20 protein AAs, within their side chains. [Also, number of hydrogen atoms within amino acid molecules appears to be in relation to natural numbers sequence; so, in side chains there are: 1, (2), 3, 4, 5, 6, 7, 8, 9, 10 of hydrogen atoms, and in whole molecules: 5, (6), 7, 8, 9, 10, 11, 12, 13, 14 (Sukhodolets, 1985) (Cf. last sentence in legend of Table 7 on page 830, in: Rako64' > 8B The principle of self-similarity extends, however, substantially broader within the amino acid (genetic) code. Moreover, this extension is present parallel with Shcherbak's principle of "analogies [of the genetic code] with quantum physics" (Shcherbak, 1994). [Details of this / #) 3 )64' > :8] Thus, in the set of protein AAs, in their side chains, there is a pattern (117/87), the 117 hydrogen atoms and 87 non- hydrogen atoms. This pattern corresponds to the pattern (107/97) where 107 is the number of all atoms in 10 polar AAs [(KR-ST-DE-NQ) + (YH)], and number 97 represents the number of atoms in 10 AAs that are not polar [(GP-AL-VI) + (CM-FW)] [The grouping of AAs follows from the distinction of AAs in four types of diversity 3 )64' > ' > 8.] In this classification calculated purely polar AAs only, and not those that are semi-polar (GP and W). Classification by the polarity criterion, however, follows from the polarity of the functional groups, that is, from the electronegativity of the atoms that make them. In view of the "chemical eyes", this polarity/nonpolarity is directly visible, and it agrees with the strict parameters for which this polarity is measured [polar requirement (Woese et al., 1966), hydropathy index (Kyte &
7 Specificity is, among other things, in the fact that AAs of alanine stereochemical type are taken by once and of non-alanine type (GP-VI) by twice (cf. Remark 3). In addition, the specificity is that "specific amino acid arrangement" exists in a direct connection with the system of amino acids precursors, with their order of inclusion, thus confirming each other - the choice of AAs and their precursors.
94 Doolittle, 1982) and cloister energy (Swanson, 1984)], as we have shown in previous works (Rako6evi4 < Jo)4' 447C Rako6evi4' >>9a, Section 3.3). A direct self-evident of polarity/nonpolarity follows from the facts as stated herein: aliphatic amino acids that in the side chains possess only electropositive atoms, C & H, or with them also possess an electropositive sulfur atom more, are nonpolar (AL-VI-CM) . [Under electropositive we consider those atoms whose electronegativity (according to Pauling) is 2.5 or less, while with values more than 2.5, the atoms are electronegative (polar).] Aromatic amino acid phenylalanine (F), which in the side chain only has a nonpolar benzene ring, is also nonpolar amino acid. On the other hand, aliphatic amino acids that in the side chains, except carbon and hydrogen (C & H), also have electronegative nitrogen atoms (N) and/or oxygen (O), are polar amino acids (KR- NQ-ST-DE); aromatic amino acid tyrosine (Y) which in the side chain has a phenol group is also a polar amino acid; Finally, the aromatic amino acid histidine (H), which in the side chain has a polar imidazole ring, is also a polar amino acid. On the third side, there are three more amino acids, that are not polar but semi-polar. First of them is glycine (G). Its molecule has two equivalent hydrogen atoms, which at the same time exist in both the "head" and the side chain of the glycine amino acid. Although the hydrogen atom itself is electropositive (nonpolar), the influence of the polar "head", i.e. the influence of the polar amino acid functional group is inevitable, and hence the semi-polarity of this amino acid is followed. Similarly, the proline has three nonpolar CH2 groups in the side chain, but one of the two ending allyl groups is attached to the polar nitrogen atom in the "head" of the amino acid; hence its semi-polarity follows. Aromatic tryptophan (W), which in the side chain contains an indole ring, composed of one nonpolar benzene and one polar pyrrole ring, must also be a semi- polar amino acid (Remark 4). It is understood that it makes sense to ask a question about the classification of protein amino acids into the class of strictly nonpolar amino acids (7 AAs) and the class of polar amino acids where semi-polar AAs also come (total 13 AAs). Such a classification shows a strict separation of the TGC into polar/nonpolar as well as inner/outer space with the differences for 1 molecule and 1 atom, respectively 3 )64' >>>' 0 ?' TI -4, pp. 279-280; 2011b, Eq. 11-14 on p. 838 and Tab. A.3 on p. 840).
Remark 4. Today, there are many different classifications of protein AAs by the criterion of their polarity. Hence, this has become a major problem not only in the analysis of protein polarities, but also in the analysis of their evolution, and it has also become a problem for understanding evolution in general. This upper representation of the polarities of AAs follows from their structure, which should become a main criterion for analyzing the structure and evolution of proteins, and also for re-examining the validity of existing parameters. In this sense, we have also demonstrated the high reliability of polar requirement (Woese et al, 1966), hydropathy index (Kyte & Doolittle, 1982) and cloister energy (Swanson, 1984). Thus, in the case of structurally semi-polar amino acids, the state of polarity/non-polarity is as follows: Pro is polar in both parameters M hydropathy index and cloister energy, but according to the polar
95 requirement it is not polarized, as Rumer and Konopeljchenko showed (1975, pp. 473-474); Gly and Trp are nonpolar after the cloister energy, but after polar requirement and hydropathy index are polar. On the other hand, in all other cases of strictly indicated polarity and non-polarity, there is the full agreement with hydropathy index and cloister energy (as well as with each other), while for polar requirement there are minimal deviations.
* All together in the question are systemic and systematic natural arrangements, whose organization and determination correspond with the principle of self-similarity.8 The already well-known facts that genetic code represents an analogy with natural (verbal) language (First Paradigm!) are joined now to the facts about analogies between genetic code arrangements and specific arrangements within the set of natural numbers (Second Paradigm?).9 Such an agreement (through "ideal" and "intelligent" structures and arrangements) leads us to the conclusion that the genetic code was indeed complete even in prebiotic conditions (as a set of 20 canonical amino acids and the set of 2+2 pyrimidine/purine canonical bases, respectively); and the "evolution" of the genetic code can only mean a degree of freedom of standard genetic code, i.e. the possible exceptions and deviations from the standard genetic code, as we have presented here and in previous works, and, as also other authors have pointed out (Sukhodoletc, 1981, 456C 0# ' 459C )64, 1988a, 1988b, 2004a; Popov, 1989; Shcherbak, 2008; Castro- Chavez, 2010). And, secondly, there is no point in talking about the evolution of genetic code in terms of the evolution of organisms, but only about the degree of freedom of a unique coherent and harmonious system, which degree of freedom is expressed in the form of exceptions and/or deviations (Remark 3) . All the present arrangements and regularities are an addition to the previously revealed systems and arrangements of protein amino acids (as constituents of GC) in which chemical distinctions and classifications are accompanied by arithmetical and algebraic regularities, both in our previous works and in the works of other authors (Rumer, 1966; Kyte & Doolittle,1982; Doolittle,1985; Sukhodolets,1985; Leunissen and De Jong,1986; Taylor & Coates, 1989; Koruga,1992; Shcherbak, 1994; Damjanovic, 1998; Dragovich and Dragovich, 2006; Négadi, 2009; Castro-E ;' > >C -ic, 2011; Petoukhov, 2016; Dragovich et al, 2017). On the other hand, all results, presented in this chapter M all together M show that the genetic code is a deeper essence than would follow only from the chemical properties of the amino acid molecules. And it also means that biological, as such, is a deeper essence than it follows only from physics and chemistry. It also proves to be a justified conclusion, at least as a hypothesis for further researches, it would make no sense to speak about the biological evolution of GC, and
8 OJ # ' E $ E )/ M only those considered to be the most important: 1. The principle of systemic self-related and self-similar ; GP 3 )64' >>9' ? 8 9 OJ ' U # M twenty amino acids and four amino bases (two pyrimidines & two purines M is involved still one "hidden alphabet", the series of natural numbers, with all its # P 3 )64' > ' 9). (Cf. footnote 3.) 96 about its degeneration, but only about the pre-biological generation of GC, the completion of its basic (2Py+2Pu bases) and acidic (20 AAs) constituents, such that their arrangements, from the aspect of physical and chemical properties, are in a correspondence with the arithmetic-algebraic arrangements of the number of molecules-atoms-nucleons in them. Existing, now known, exemptions from this generic standard GC are only necessary deviations as a result of the degree of freedom that is inevitably valid for all natural systems.
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97 Petoukhov, S. (2016) The system-resonance approach in modeling genetic structures, BioSystems, 139, 1M11. Popov, E. M. (1989) Strukturnaya organizaciya belkov. Nauka, Moscow (in Russian). )64' - - 3 4558 -dimensional model of the genetic code, Acta Biologiae et Medicinae Experimentalis, (Prishtina), 13, 109-116 [An excerpt in: http://www.rakocevcode.rs).] )64' - - 3 4558 $ ' - ' . 3 0 # T . 0 8' +6 )&' ( 3http://www.rakocevcode.rs)
)64' -M. (1991) The coherence of the chemical and genetic code, Proceedings of Faculty of science (former: Faculty of phylosophy), Chemistry Section, 2, 1-4' + )64' -& 3-)8 3 4498 Logic of the Genetic Code' +6 )&' ( (http://www.rakocevcode.rs )64' - - 3 44:8 # /-tRNA synthetases in correspondence with the Codon path cube. Bull. Math. Biol. 59, 645-648. )64' -- 3 44:8 The genetic code as a unique system' 0 ) ) ' + (www.rakocevcode.rs) )64' -- 3 4458 $ / ' Biosystems, 46, 283-291. )64' - - 3>>>8 sification of protein amino acids, Proceedings, (Glasnik) of the Section of Natural Sciences of Montenegrin Academy of Sciences and art (CANU), 13, 273-294. (arXiv:q-bio/0611004 [q-bio.BM])arXiv:q-bio/0703011v2 [q- bio.OT]. )64' -- 3>>98 c structure of the genetic code, Journal of Theoretical Biology, 229, 221M 234. )64' -- 3>>98 D $; $ - TQ "Divine" Equation, FME Transactions (FME = Faculty of Mechanical Engineering, Belgrade, Serbia), 32, 95M98 (also in: arXiv:math/0611095 [math.GM]). )64' - - 3>>78 $ / ' X=I-bio/0610044v1 [q- bio.OT]. )64' - - 3>>:8 # ' X=IF>:>?> AI-bio.GN]. )64' -- 3> 8 Genetic Code: Four Diversity Types of Protein Amino Acids, arXiv:1107.1998v2 [q-bio.OT]. )64' -- 3> 8 $ / Neuroquantology 9 (4), 821M841. )64' - - 3> ?8 [ e, arXiv:1305.5103v4 [q-bio.OT]. )64' - - 3> 68 T "# " www.rakocevcode.rs stored on 2015-01-06.
98 )64' - - 3> :8 $ E E www.rakocevcode.rs (stored also at "OSF Preprintson" 2017-08-09 and new, minimally modified, version 2017- 10-02). )64' --' *)4' 3 4478 D / = / determination with chemical characteristics, atom and nucleon number. J. Theor. Biol. 183, 345M349. Rumer, Yu, B. (1966) O sistematizacii kodonov v geneticheskom kode, Doklady Akad. Nauk. SSSR, 167, 1393-1394. Shcherbak, V.I. (1994) Sixty-four triplets and 20 canonical amino acids of the genetic code: the arithmetical regularities. Part II, J. Theor. Biol., 166, 475-477. Shcherbak, V. I. (2008) The arithmetical origin of the genetic code, in: M. Barbieri (ed.), The codes of life: the rules of macroevolution (pp. 153-181), Springer, Berlin. Spinadel, V.W. de (1998) The Metallic Means and design, In: NEXUS II- Architecture and Mathematics, Ed: Kim Williams. Spinadel, V.W. de (1999) The family of metallic means, Visual Mathematics, 1 (3), 1-16, http//members.tripod.com/vismath1/spinadel/. Sukhodolets, V. V. (1981) Evolucionie preobrazovaniya geneticheskogo koda, predskazivanie iskhodya iz gipotezy o fizicheskom predopredelenii naborov osnovaniy v sostave kodonov, Genetika, XVIII, 3, 499-501. Sukhodolec, V.V. (1985) The meaning of the genetic code: the reconstruction of the stages of prebiological evolution (in Russian), I , XXI, 10, 1589 M 1599.] Swanson, R. (1984) A unifying concept for the amino acid code, Bull. Math. Biol. 46, 187-207. Taylor, R.J.F., Coates, D. (1989) The code within codons. Biosystems 22, 177M187. Weaver, R.F. (2012) Molecular Biology, fifth edition, McGraw-Hill, New York. Wetzel, R. (1995) Evolution of the Aminoacyl-tRNA Synthetases and the Origin of the Genetic Code. J. Mol. Evol., 40, 545-550. Woese, C.R., et al. (1966) On the fundamental nature and evolution of the genetic code. In: Cold Spring Harbor Symp. Quant. Biol., 31, 723-736. Wohlin, Åsa (2015) Numeral series hidden in the distribution of atomic mass of amino acids to codon domains in the genetic code, J. Theor. Biol. 369, 95M109.
99 100 Chapter 4
The Enigma of Darwin's Diagram
What can be said at all can be said clearly, and what we cannot talk about we must pass over in silence. Ludwig Wittgenstein ("Tractatus")
According to my best knowledge, for the first time here is presented a hypothesis, that the one and only "accompanying diagram" in Darwin's famous book On the Origin of Species contains, may be, a hidden code. Direct inspection reveals that the Diagram, viewed as built of four parts [(two upper and two lower / two left and two right); (two with more and two with less branches / two with multiple and two with single branches)], corresponds to the logical square of the genetic code. When, however, viewed as built of two parts (upper and lower), then it corresponds with Shch)R 30 )' 44?' 4498 -codon and non-four-codon amino acids (AAs); not only by the form but also by the number of elementary quantities. The number 0 )R 3-codon amino acids) is determined by the Pythagorean law (3^2 + 4^2 = 5^2 = 25), meaning that the total number of nucleons makes the product of the number 25 and "Prime quantum 037" (925); and the number of branches in the # "#R / # Plato (3^3 + 4^3 + 5^3 = 6^3 = 216), meaning that the total number of branches makes the product of the number 216 and "First I > @ 3 78 G ' # 0 )R 7> of "Prime quantum 037" (2220), wh "#R 7> "First quantum 01" (60). There are 216 + 60 = 276 branches (in total), and this number is also the number taken from a specific and unique arithmetical system. Furthermore, it is shown that Darwin, starting from the basic structure of the Diagram, formed a sophisticated structure which strictly corresponds to the arithmetical and /or algebraic structures that also appear to be the key determinants of the genetic code (GC). Among other correspondences, there is also one in the FI ' # 0 )R within the amino acid constituents of GC (in their side chains) is as follows: [1 x (G1+A15+ P41+ V43+ T45 + C47 + I57+ N58 + D59 + K72 + Q72 + E73 + M75 + H81 + F91 + Y107 + W130)] + [2 x (S31 + L57 +R100)] = 1443. J 0 )R ' # more, for the number of atoms, the result is as follows: [2 x (G1 + A4 + C5 + D7 + N8 + T8 + P8 + E10 + V10 + Q11 + M11 + H11 + I13 + F14 + Y15 +K15 + W18)] + [3 x (S5 + L13 + R17)] = 0443. G ' # "#R U @ @ entities/quantities: 276 branches, plus 46 nodes, plus 10 branchings, in total 332. The significant differences are as follows: 1443-332 = 1111and 443-332 = 111, both determined by the unity change law. From these results it follows that Darwin with his Diagram anticipated the relationships not only in terrestrial code but in the genetic code as well, anywhere in the universe, under conditions of the presence of water, ammonia and methane, phosphine and hydrogen sulfide. If so, then Darwinian selection moves one step backwards in prebiotic conditions, where it refers to the choice of the life itself. 101 1. Introduction
As it is generally known, Darwin's book On the Origin of Species contains only a single illustration, an evolutionary tree in the form of a diagram (Figure 1.1). During the 155 years since the appearance of the first edition in 1859, this Diagram has been analyzed only qualitatively (Figure 1.2), but not quantitatively, and we shall, in this chapter, do that for the first time.1 In doing so, we begin with the working hypothesis (for this and all other researches of the Diagram in future) that the diagram contains a hidden code, with strictly determined quantities, expressed in the number of branches M primary (principal, main) and secondary (minor, small)2, and also in the number of nodes and branchings; such a code, which would per se have to be biological, otherwise it would not make sense in this book, and the Diagram would not be styled as "accompanying diagram" but as an "attached diagram", or an ordinary illustration. Hence, the deeper implication of the hypothesis is that, despite the variations (and modifications) of organisms are spontaneous and random, they do not have complete freedom, but are limited by the regularity and validity of strict arithmetical and/or algebraic systems. (Cf. Box 1.)
Box 1. Citation from 1994 (I) )64' 449' 9= O"#R Mbinary tree, represents the first systematic informational approach to the analysis of the relations between organisms. This is the only diagram in his book Origin of Species (Darwin, 1859) and it represents a model for interpretations of origin of varieties, species, genera and higher systematic categories. By its essence, this diagram represents a code-model and code-system and by its completness and complexity it is the first example of the code model and the code system in science. Relations of the elements within this code system correspond to the relations of the elements (organisms) in natural systems. Intention (and a message) of the author of this diagram is absolutely clear: if the natural systems are at the same time the coding systems, the only adequate and complete way of description and interpretation of such systems would be the creation of adequate code models with adequately corresponding relations between the elements of one and the other model, i.e. / O
The analysis that we conducted showed that the relationships between these quantities are such that they are brought into mutual relationships by specific proportionalities and balances through the minimal differences in number, usually expressed in decimal units (± 00, ± 01, ± 10, ± 11, ± 100, ± 111 and so on)3, with the validity of the principle of minimal change, and the
1 In fact, this is the third time. The first time, it was twenty-three 3 )64' 44 8, and the second time, it was 20 years ago 3 )64' 1994). But both times it was only a pilot study, which was to serve as the initial "trigger" for a comprehensive analysis, the results of which are now presented. 3 )64' 44 ' 9= ` / O8 3 )64' 449' 7= (U (U 8 2 Primary branches go from the previous level (line) and they always reach the next level (and they are designated by letters). Secondary branches, however, fail to reach the next level, they are not finalized; they do not become a taxonomic category (a variety, species, and so on.) 3 O0 /' / / ' ' ystem is the P 30 )' >>5' 6:8
102 principle of continuity.4 Moreover, all of these quantities were related and corresponding to the quantities (and their relationships) in the genetic code; with the number of codons, molecules, atoms, nucleons etc. The obvious reason why this is so, is (according to our working hypothesis) the fact that Darwin in his Diagram built relationships taken from the specific and unique arithmetical and/or algebraic systems, based on which, as we now know, the genetic code was also built.
2. Methodology
Bearing in mind that the genetic code is the basic biological code, and that it has already been proven that its distinctions and classifications (within itself), are derived on the basis of physico- chemical properties of the molecule, followed by (accompanied by) strict arithmetical and/or algebraic regularities and balances (Shcherbak, 1993, 1994, 2008; Damj4' 445' >>6' >>7C %) ' 449C " ' >>4' > C -4' > C + ' >>4' > 9C E - Chavez, 2010, 2011; Dlyasin, 2011; Joki4' 447C )64' 44:' 445' >>9' > ' > ?8' makes sense, in analysis of the distinction and classification in Darwin's diagram, to apply the same methodology (or almost the same) by which the said regularities in the genetic code were discovered. This means that the number of branches, nodes and branchings must be determined in even and odd positions; along cross diagonals, and zigzag lines; for different parts of the Diagram, which basically boils down to the application of Mendeleevian methodology, that can be found in his original manuscript works (Kedrov, 1977). B.M. Kedrov, who most carefully studied the archives of Mendeleev, said that he was unable to find that Mendeleev wrote about which methodology he had used in his researches. In contrast to this, handwritten sketches, drawings and diagrams show that Mendeleev clearly revealed his methodology. In the above mentioned book, Kedrov enclosed 16 photocopies (between 128 and 129 pages)5, showing the Mendeleevian methodology; which is the same methodology as we applied in the analysis of the genetic code structure as well as in the analysis of Darwin's diagram.
4 [ # - R C But we also bear in mind the validity of these two principles in the genetic code (Swanson, 1984, p. 187). 5 All of these copies, plus two tables, can be found on our website ("The Mendeleev's archive"). Those particularly significant are: a copy (copy I, p. 128) which demonstrates "the chemical patience (solitaire)"; copy IV, which presents the chemical elements in the even/odd positions, with a drawing which indicates the number of odd and even valences, and the atomic mass differences are presented using the Pythagorean method of determining the differences in tetraktis (by Mendeleev in n-aktis); and copy VIII with the diagonal relations drawn in the Periodic system table.
103 3. Preliminaries Already at first glance, it becomes immediately obvious that Darwin's diagram (Figure 1.1), composed of four parts (two upper and two lower / two left and two right); (two with more and two with less branches / two with multiple and two with single branches), corresponds to the logical square of the genetic code, in a reverse reading6 (Figure 2), as well as with Shcherbak's diagram at the same time (Figure 3), also in the reverse reading.7 Two lower trees are branched, multiple, and two on the top are linear, non-branched, with linear segments. In the lower left part of the Diagram, the tree consists of two large branches, and the tree on the right consists of only one. In the upper, left part of the Diagram, there are more singlet branches (eight), and on the right there are less branches (six).8 # 0 )R # = @ @ molecules have the same number of nucleons each, and their bodies are completely different. It is (by analogy) similar to the Darwin's diagram: the singlet branches are implemented in the same number at every level, and the multiple branches in different number, changing from level to level. But it is so at first glance. However, the second (deeper) look reveals a surprising fact: the 0 )R determined by the Pythagorean law (3^2 + 4^2 =5^2 = 25), meaning that it is 25 of "Prime quantum 037" (925), and the number of branches in the lower part of Darwin's diagram is determined by the law of Plato (3^3 + 4^3 +5^3 = 6^3 = 216), meaning that the amount is 216 of @D I > @ 3 78 G ' # 0 )R gram 7> @H I >?:@ 3>8' # "#R there are 60 of "First quantum 01" (60).9 [A total of nucleons is 925 + 2220 = 3145, and a total of branches is 216 + 60 = 276, which is again a number taken from a specific and unique arithmetical system, as the first case (Figure 4).] [Remark 3.1. J # ) 0 )R 3 Shcherbak, 1994): 037, 370, 703, it is clear that the first two steps can be realized by all two-digit numbers, while the third step (through module 9) is possible only for number 037; for example (037, 370, 703) versus (038, 380, 722).]
6 Positioning "from smaller to larger" in the genetic code is from the left to the right, and in Darwin's diagram it is from the right to the left. 7 J 0 )R upper part of the Diagram and the large part is down in the lower part of the Diagram, while in Darwin's diagram it is the opposite. However, as the first inversion (with respect to the genetic code) is essentially natural, the latter is completely random. 8 This "first glance" refers to descendants that follow from the species "A" and "I", whereas for the remaining species (B, C, D, E, F on the left and G, H, K, L on the right), the situation is somewhat different, and that will be explained in the text which follows. 9 All branches (the sum 60 + 216 = 276) which are the descendants of all 11 species designated with large Latin letters at the bottom of the Diagram are included into this counting.
104 "#R ; 3 8 9 ' / Roman numerals. At the bottom of the Diagram, there are 11 English alphabet letters, A-L,10 ommiting the 10th letter (the letter "J").11 Because of this exclusion, the original input order: J- 10, K-11, L-12, (M-13) becomes the output of order K-10, L-11, (M-12).12 In support to the assumption that here the term of coding is already present, there is the fact that the branches are omitted only at the 10th level.13 On the other hand, it is also a fact that the omission of capital letters begins with "M" (the 13th, central letter in the English alphabet), and alignment of small letters on the second branch of the left tree begins (and continues) exactly with "m". In addition, only the levels 11, 12 and 13 are not marked with small letters, while all the others are. The omitting of the 10th letter makes another distinction: only the letters after the 10th letter are put into a new sequence, they are "variable". However, the letters from the 1st to the 9th remain unchanged, they are "stable". From that fact it follows that the main part of the Diagram is bounded by the first and by the last stable letter, "A" and "I". The species of organisms that are designated with these letters differ in other formal characteristics. Hence, we can speak about two sets of species: the first set of two, and the second set of "other nine species". In the first set of species, the branches (below the 10th level) are oblique (oblique angle), while in the second set the branches are orthogonal; within the first set there are nodes and branchings whereas within the other set there are not. By this, both types of branches (oblique and orthogonal) exist in both parts of the Diagram, in the left part, A-F, and in the right part, G-L. The above reconciliation: 10th letter vs 10th level; "M" vs "m"; significant omission of capital letters at the start level versus reordering of the 11th, 12th and 13th letters (K, L, M), as opposed to the exclusion of small letters at the top of the Diagram at the positions 11th, 12th and 13th; all these relationships represent a kind of the specific realization of similarity principle and the principle of self-symilarity.14
10 J "#R # = ` / # s in understanding this rather perplexing subject. Let A to L represent the species of a genus large in its own country. These species are supposed to replase each other at unequal degrees, as is so generally the case in nature, and as is representes in the diagram by the letters standing at I O 3The Origin of species, 1876, Chapter IV, p. 90). 11 One might think that this omission is done because the two adjacent letters "I" and "J" are similar to each other, so that Darwin wanted to avoid confusion. We, however, believe that this is such a code, which requires the omission of only the 10th letter, no matter how it looks. 12 As if Darwin wanted to tell us something about these numbers; perhaps to present their uniqueness: [(11/11, 22/22, 33F??' G ' 44F448' 3 F ' 9F9' ?7F7?8' 3 ?F? ' 7F7' ?4F4?8B 3 U 8 13 This absence of branches should not be confused with the fact that at every level the branches (taxonomic entities) from the previous level are finalized, so thus, branches whose development started at the 9th level are finalized at the tenth level. 14 Future researches should show whether this self-similarity is of fractal and/or non-fractal nature. A significant fact with regard to this, is Darwin's insisting on the fact that the structure of the Diagram can also refer to various taxonomic categories. (The Origin of species, Chapter IV, p. 91: "When a dotted line reaches one of the horizontal lines, and is there marked by a small numbered letter, a sufficient amount of variation is supposed to have been accumulated to form it into a fairly well-marked variety, such as would be thought worthy of record in a systematic #)PC E XJ' ?>?= @M / ' and the dotted 105 4. Results and discussion 4.1. Primary and secondary branches of species "A" and "I" In our working hypothesis, there is a presumption that the symmetry relationships make the basis for coding, and for that reason we have analyzed the number and arrangement of branches, nodes and branchings on the 15 levels of the Diagram, at first, in symmetrical systems "2 x 5" and "3 x 5", and then in systems derived from them. Such symmetrical systems are presented first in Table 1.1, Table 1.2 and Table 2.1. The number of primary (main) branches on the left tree (starting with letter "a") and the right tree (starting with the ending letter "z"), for the species "A" and "I" is given in Table 1.1.. The branches are counted starting from the zeroth level onwards, until the ninth, by counting the number of branches between every two levels. The same result is, however, obtained when we follow the finalization (realization) of taxonomic entities at every next level (Table 1.2). In the latter case, we start counting with the first instead of the zeroth level and we end counting with the tenth instead of the ninth level (by this counting we realize that the number of branches is equal to the number of letters per level). From the aspect of this vision, all primary branches are "finalized" (and marked with the corresponding small letters at the lower part of the Diagram and the unmarked ones are in the upper part of the Diagram); they are further classified into two classes: 1. Finalized, fixed (Table 1.3), and 2. Finalized, not-fixed (Tables 1.4 and 1.5). These first branches reach a certain level and do not develop further; as examples, we show the first such branch on the left tree (s2), and the first such branch on the right (t3). If we take any of the two tables (Table 1.1 and Table 1.2) and look at the upper half of the large (left) and lower half of the small (right) tree (and vice versa), then, in this cross-connection, the number of branches is equal (28 and 28).15 But apart from these symmetrical proportionalities to the total number of primary branches (28:28 = 1: 1), there is one more such proportionality valid for the parts of the system (20:20 = 1: 1) (the total number of primary branches on the small tree equals the number of branches on the upper half of the large tree);16 and there are also the following proportionalities: (36:24 = 3: 2), (32:24 = 4: 3), (8: 16: 24: 32 = 1: 2: 3: 4) etc. In Table 2.1 we look at all primary branches, up to the 14th level. However, prior to the analysis, an important issue should be considered. In fact, according to the said first counting procedure, on the tenth level there are no branches; according to the other procedure, however,
G ; / ' / UP8 motif extended along the overall evolutionary lines. 15 Is it just a curiosity, that number 28 is the second perfect number? 16 The same or similar proportionalities exist for the number of nodes, as well as for the number of branchings, which will be discussed further.
106 we say that on the tenth level, three branches on the left, and two branches on the right tree (which arrived from the previous ninth level) are finalized. Then, the question is whether, in this second sense, there are also branches (descendants) at the eleventh level? The answer was given by Darwin himself,17 from which it follows that all four levels of the upper part of the Diagram contain finalized branches, which arrived from the previous 10th level: 8 on the left and 6 on the right.18 The first thing we see in Table 2.1 is that the number of branches in the upper part of the Diagram is equal to the number of branches in the lower part of the Diagram (56 + 56 = 4 x 28 = 112); then, that the result of cross-linking system components (along the two zig-zag lines), the pattern 52/60, as well as the total number of branches (112), was taken from a specific and unique arithmetical system (Fig. 5). In addition, this number of branches (112) is just a permutation of the number 121 (11^2),19 which is actually the number of secondary branches on both trees, for the two species, "A" and "I" (Table 2.2)20; and this number is also taken from a specific and unique arithmetical system, which we have already presented in the Preliminaries (Figure 4). Figure 4 shows several things at the same time. First, it presents a clear and unequivocal arithmetical system which from, as we have seen, Darwin took (reconciled) the results for the total number of branches in the Diagram (276) as well as for the number of secondary branches from zero up to the 9th level of the Diagram, the number 121, for the species "A" and "I" (Table 2.2). But at the same time we see that these results follow from the determination by the first perfect number, the number 6, which also appears to be the determinant of the genetic code (Figure B.2).21
[Remark 4.1. Secondary branches do not have branchings, while the primary branches have. As examples, the two positions at the first level on the left tree: from a1 there is not, while from m1 there is a branching; details about speaking in Section 4.4, in tables 3.1 - 3.3 (the nodes and branchings), in relation to tables 4-1 - 4-5, where there are the sums of the primary and secondary branches.]
17 The Origin of species' 5:7' E J%' 4 = OJ -thousandth generation, and under a condensed and simplified form up to the fourteen- P 18 The Origin of species, 1876, Chapter IV, p. 94: "Hence very few of the original species will have transmitted offspring to the fourteen-thousandth generation. We may suppose that only one (F), of the two species which were least closely related to the other nine original species, has transmitted descendants to this late stage of descent. The # ' # # P 19 Notice that square of 11 (112 = 121) is zeroth case in logical-arithmetical arrangement presented in Table A.1; also, the tenth part of the fourth friendly number, 1210 [more exactly, the second member of the second pair (1184 & 1210) of friendly numbers]. 20 J ' O @ ' >F> ' /= The pattern 52/60, valid for all primary branches (Table 2.1) was changed into the pattern 62/59 (Table 2.2), valid for secondary branches (cf. Section 4.6, first paragraph). 21 More details on the determination of GC by perfect and friendly numbers see in Ra)64' 44:' 7>
107 4.2. The riddle of the genetic code Table 2.2 is very significant. It is amazing that the sequence of quantities: 11, 22, 33, 44, 55, 66, 77 is realized.22 It is hard to believe that it could be a coincidence, especially if we know that just by these numbers a specific and unique arithmetical system, which is one of the most ' 3 E U E8 3 )64' 2011a, Table 4; 2011b, Table 4). The understanding of that determination is easier by illustrations given in Appendix C, where it is shown that the said arithmetical system contains the specific algebraic system, which also appears to be a significant determinant of the genetic code: it determines codon/amino acids assignment in relation to a classification into four diversity types of amino acids (AAs). In Figure C.1 the classification into four diversity types is shown, in linear and circular form; and Figure C.2 shows the manner in which the circular arrangement becomes a Table of Mendeleevian type, where the molecules are arranged, mutatis mutandis, in accordance with the principles of minimum change and continuity. But what is surprising is the fact that the quantities (26, 42, 57, 77), representing the number of atoms in this Table (Figure C.2) are "taken" from the arithmetical system, given in Table C.1 (in relation to Table C.2 and C.3), in a manner as shown in Survey C.1. According to the algebraic equations given in Survey C.2, the 25 codons encode for less complex, and 36 for more complex AAs (Table C.4).
4.3. Darwin's solution to the riddle of the genetic code The missing link in the strict determination of the genetic code by an arithmetical (Table C.1) and an algebraic system (Survey C.2 in relation to Survey C.1) is actually in the Survey C.2. In fact, we do not know which quadruplet sequence is preceded by or which one follows a sequence of squares (6^2, 5^2, 4^2, 3^2); moreover, we do not know which sequence is initial, and if there is a more general law that all the sequences are connected with? Fortunately, there is an answer, and it is contained in Darwin's diagram (Figure 6 & 7 in relation to Tables 5, 6.1 and 7.1).23 # / ' [ )R + ^ 39 + 2) (n = 0, 1, 2, 3 ...), # ' + 4 stable aromatic molecules; and by analogy, the number of chemical elements in the periods of the periodic system of Mendeleyev (2s, 6p, 10d, 14f ...).24 (Cf. Box 2.)
22 Table 4.5 presents the missing 88 (all branches on the second tree, for the "A" and "I" species, in 3 x 5 arrangement, 0-14 level), and again Table 7.5 (primary branches in all 11 species , 0-9 level); in Table 5 there is the number 99, also missing in this sequence. 23 In relation to Table 6.1 there are Tables 6.2, 6.3, 6.4 and 6.5, in relation to Table 7.1 there are Tables 7.2, 7.3, 7.4 and 7.5. 24 # # # + ^ 3Z 8 3 ^ >' ''?8 O P ' form N = (2n+1) (n = 0,1,2,3) is just a formula for calculation of the odd numbers and the number of atom orbitals: ' ?' 6 ' : G
108 Box 2. Citation from 1994 (II) )64' 449' 9= O ' # Mbinary tree, is the realization of the logic of the systematization and classification, separation of the parts within the whole, as well as the regularity of the hierarchy of the levels. The accordance of this logic with the model of classification of the number systems with the + ^ 3Z 8 3 ^ >' ''?8 / G 0' # ^ >' + = 2, what corresponds to the division of binary tree to the left tree and the right tree. It "#R / # # Q 38R Q 3J8R G J case when n = 1, N = 6, and this again corresponds to the division of the tree, to the left and right tree, but in this case this division is strictly indicated by only one line, the line of the letter (species) F which has a positional value of exactly 6 (this is the sixth letter in 8 G U / the level hierarchy is the case when n = 2 and N = 10. This situation corresponds to a reduction of all branch outputs to three and two outputs [on the 10th level] on the left and right tree ... In the latter case, n = 3 and N = 14, what corresponds to the end-outputs of 3 9 8 # Q# ' 38RC Q 3J8 # / U # R25
By this rule, as we now see, the connection between the quadruplets of squares is determined, in a series of natural numbers, through a system of two and two linear equations,26 which are connected by an "inserted" intermedial equation. In the case of the genetic code these three equations are found in the third "quadrant" of the system in Figure 7 (correspondingly with Survey C.1 and C.2, as well as Table C.4), with the intermedial equation as Darwin's equation (27 + 09 = 36), which is found in Table 5 and Table 6.1; it determines the number of primary branches in the "9 other species" (out of species "A" and "I"). Hückel's rule (more precisely, an analogue of the rule) is a generalization concerning the "travel" of quadruplet squares generated from a series of natural numbers, starting with quadruplet 1-2-3-4, that is with 1^2M2^2M3^2M4^2. But knowing now for this Darwin's generalization that contains Hückel's rule, (and is related to the squares), as well as for Darwin's Platonian solution, given in the Preliminaries, and it concerns cubes, a new question is: Is a generalization over the n-th degree possible (n = 1,2,3,4,5 ...)? In our opinion, the answer to this
25 In addition to what was written 20 years ago, now some refinements are given. It is obvious that Darwin in several different ways makes distinctions corresponding to the Hückel's rule. Two ways are explicit, one in a set of letters, and another in the set of the branches. First, we present solutions in the set of letters. So, the case for n = 0, and N = 2 refers to the second letter of the alphabet (B), which begins the second set of species. [In the first set there are (A, I), while in the second set there are (B, C, D, E, F, G, H, K, L).] The case for n = 1 and N = 6, refers to the 6th letter (F), which separates the left tree from the right tree in the Diagram. The case for n = 2 and N = 10 refers to the 10th letter (J), which is excluded. The case for n = 3 and N = 14, refers to the 14th letter (n), which for the first and for the last time appears on the 14th level. [Letter n as 13th, the middle letter reading backwards.] The solutions in the set of branches are these: on the 2nd level, a first fixed branch appears (s2); after the 6 th level there is no branching; on the 10 th level there is the finalization of the branches from the lower part of the Diagram, and on the 14 th level there is the finalization of the branches from the upper part of the Diagram. 26 Two linear equations whose unknown quantities are linked with a plus sign and two are associated with a minus sign.
109 question should include the - R I ' `" #) [/ + ) O AO(; ; ) S beiden Stammflanzen, so gibt 3n (3^n) die Gliederzahl der Kombinationsreihe, 4n (4^n) die Anzahl der Individuen, welche in die reihe gehören, und 2n (2^n) die Zahl der Verbindungen, # ) OB27
4.4. Nodes and branchings Now we observe the Diagram (Figure 1.1) compared to Table 3.1. At the zeroth level we find a node on the left tree as well as on the right tree. At the first level, there are two nodes on the left and one node on the right etc., until the ninth level, after which there is no node involvement. Some nodes branch and some do not. By this, one must notice that there is a branching only when one of the nodes is followed by at least two branches, which are finalized at the next level (and they are marked by letters). Thus, the node at the zeroth level on the left tree is at the same time a branching, while on the right it is not (Tables 3.2 and 3.3). It is easily seen that after the sixth level there is no more branching. [On the sixth level there are the following branchings: m6 branches into m7 and l7 on the left; z6 branches into z7 and w7 on the right.] This fact requires that, in the analysis of the number of all branches, except the splitting into the 5 + 5 levels as in Table 4.1 we must analyze the splitting into 7 + 3 levels28 as in Table 4.2, and then into the 3 + 4 + 3 levels as in Table 4.3; and into 3 +2 +2 +3 levels as in Table 4.4. The analysis shows that the number of nodes, as well as the number of branchings, along the two diagonal lines, is balanced through changes by ±0 or ±1. Thus, the number of nodes is 23±1 (Table 3.1), and the number of branchings is 5±1 in Table 3.2 and 5±0 in Table 3.3. The same balances were carried out in the odd/even positions. The essential connection of nodes and branchings allows the possibility of their addition: 46 nodes + 10 branchings equals 56 group tree-entities (Tables 3.1 and 3.2) in correspondence with 56 primary branches as individual tree-entities, both in the lower and in the upper part of the Diagram (Table 1.1 and 1.2 in relation to Table 1.5).] That essential connection is related to the fact that both primary and secondary branches spring from the same nodes (Table 3.1). But what is "unacceptable" concerning the addition is that some nodes (the ones in which there is a 8 # [#' ) O P # ind # # 0 )R (cf. legend to Figure 3).
27 OAccording to Mendel, such system is determined by the four entities, 1n M 2n M 3n M 4n (n = 1, 2, 3 ...): Stammarten M Konstante Formen M Glieder M Individuen ... Note that Mendel only uses the term Stammarten, i.e. Stammpflanzen for the first entity but not the mathematical expression 1n # # U - R O 3 )64' 449' :78 28 However, by branching, not only levels are classified into 7 + 3, but that was also done through the distribution of branchings on the left and the right tree; on the left tree the 7 of them, and on the right 3 branchings. 110 4.5. BinaryMcodeMtree in DarwinR Diagram If we exclude (in the part of Darwin's diagram which is generated from the root "A") the nodes without branching, then we, mutatis mutandis, obtain the source Darwin's diagram (Figure 1.3). And if all secondary branches are excluded from this source Diagram, and only two primary branches are left at each node we get a "clean" binary tree, which one hundred percent / 3D )64' 445' 598
[Remark 4.2. Darwin diagrams in Figures 1.3, 1.4 and 1.5 preceded to the book The Origin of Species; first two (Figures 1.3 & 1.4) as singlet ilustrations, while the third (Figure 1.5) as a set of illustrations M a unifying set of four diagrams. (For details see: Fleming, 2013.) But, what is important for us here is the fact that the Diagram in Figure 1.5 Darwin made by hand, and in it there is a small letter "j", but as a large letter does not appear; also, there is a large letter "M" too, which does not exist in Figure 1.1. Altogether is in favor of our hypothesis that Darwin on his diagrams dealing with three sets of letters: 1. A & I; 2. B, C, D, E, F; G, H, K, L and 3. M, N, O, P, ..., Z.] And, as on the binary tree of the genetic code where there is only one possible alternative in ' "#R / / / ' s well. One by one, along a binary tree, in a very long evolutionary path, from generation to generation, the totality of alternatives (changes and modifications implemented through the process of selection) dismisses the great antinomy of the diversity of organisms (Box 3), the basis of which is the antinomy of the genetic code (Box 4). In other words, variations and modifications, which Darwin's text presents, cannot be arbitrary, but are determined and bounded by a specific and unique arithmetical and/or algebraic structures /systems, the basis of which are the following principles: the principle of symmetry, the principle of the minimal change and the continuity principle.
4.6. The balances of the number of branches for two species ("A" and "I") The number of primary branches for two species, "A" and "I", at all levels (I-XIV) is given by the pattern 52 + 60 = 112 (Table 2.1)29, which appears to be the middle case in a specific arithmetical system (Figure 5). On the other hand, the total number of secondary branches (from the zeroth to the ninth level) is such that it represents the change in 10/01 in relation to the number of primary branches, respectively: 52/60 in Table 2.1 is changed to 62/59 in Table 2.2 (52 +10 = 62 and 60-01 = 59). But what is rather surprising is that the unit balances continue ' / # / # "#R diagram.
29 Cf. Section 4.1, paragraph 6, the first to the last.
111 Box 3. "Irreconcilable" antinomy of organism equality and diversity .A. Timiryazev, Istoricheskij metod v biologii, Akademiya nauk USSR, 1942, Moscow, p. 187-188: "If all organisms are related by the unity of origin (as it is proven by general observation derived from a comparison of fact classification, metamorphosis, comparative anatomy, embryology, paleontology), then the organic world [as opposed to the vast diversity] must be a merged, inseparable whole. That sharp contrast, that irreconcilable antinomy nobody managed to resolve neither before nor after Darwin. And he himself used to stop at it, until he found a solution that, logically, followed from the same principle - the principle of selection ... Natural selection provides a better chance of survival to those beings who possess some characteristics which ensure their survival under given conditions. Among such characteristics, there is some degree of difference in relation to the other closest beings and it saves them from the competition and provides, so to speak, some space for the newcomer. Thus, a differentiation, a certain degree of difference will be useful, it will mean the success of those forms which are the most different from their parents and from each other. Darwin called this the principle of characteristic divergence (divergence of characters) and he explained it by the following scheme (Figure 15 on p. 188)" (here: Figure 1.3).
Box 4. "Irreconcilable" antinomy of the genetic code constituents equality and diversity The genetic code antinomy can be expressed in several ways, out of which we here present only two. The first way is 0 )R 3D ?8= M 6 @ @ 6 - four-codon AAs there is the same number of nucleons, as in their 15 completely different bodies (1110). On the other hand, the number of nucleons within eight four-codon AAs M in different bodies, identical heads and whole molecules M is such as to comply with the law of Pythagoras (squares of numbers 3, 4 and 5, multiplied by the "Prime Quantum 037", respectively). Despite the fact that 19 out of 20 canonical AAs are derivatives of the same AA (glycine), they build a huge number of different proteins; and the four nucleotide bases, which are derivatives of the same molecule (pyrimidine), build a number of different and various DNA/RNA macromolecules, genes and genomes.
Thus, the total number of branches (primary + secondary branches, in the classification into 5+5 levels), shown in Table 4.1, along the two diagonal lines is such that it constitutes a change of ± 01 compared to the arithmetic mean, i.e. compared to the value of the central pair of numbers: the result 90/87 in relation to 89/88. In the next step (primary + secondary, in the : Z ? 8 # 9' / / > 4>F5: 80/9730 is realized. In the next step (primary + secondary branches, in the classification into 3 + 4 + 3) as shown in Table 4.3, the arithmetic mean, i.e. the central pair of numbers (88/89) is realized. Classifications and distinctions in Tables 4.1M4.4 do not affect the number of branches at even and odd positions, respectively, which is 82/95;31 but in the fourth step (Table 4.4), in the
30 As a result of splitting the arrangement 5+5 into 7+3, a specific self-similarity also appears through the patterns (46/44 versus 66/64) in Tables 4.1 and 4.2, respectively. 31 The change of ± 02 is in relation to the diagonal result 80/97 in Table 4.2.
112 # ;; ' / > U/ 35F46 83/94). The fifth step is associated with a number of branches, from the upper part of the Diagram as well (arrangement 5 + 5 + 5) (Table 4.5), and the result of the two zigzag lines / > 3 7F : 9 F 48
4.7. The "Prime Quantum 037" It is clear, from the results presented so far, that the key principle of classification is actually a (symmetric) distinction of the system, a splitting into two parts, in proportion 1:1 (5:5). Concerning the distinction 7:3, however, there must be some additional (hidden?)32 reason; maybe the appearance of the "Prime quantum 037" or a connection to Lucas's sequence (Figure D.1), or something else? But whatever it may be, the analysis of quantitative relations in the Diagram shows that precisely this distinction (Table 4.2), with the sub-distinction 3:4:3 (Table 4.3) is the most significant. Taken together, in unity, they show that the quantities are chosen in such a way that in the final result (along the diagonal lines) they represent the realization of 3rd, 2nd and 1st OH I >?:P -' / # 3 sub-distinction in Table 4.3) that the "Prime Quantum 037" is a part of a broader arithmetical system (Table B.1 and Survey B.1)33 what we have also presented in several previous works, # # & 3 )64' >>5' ?8 J /' #/' @H V >?:@ - R calculations.34 At this point Mendeleev calculates the differences of atomic masses of elements, and in three cases makes two "mistakes". Instead of writing 30/27/67, what is actually the result, he writes 30/37/77 (Appendix B, Survey B.4).
4.8. Primary and secondary branches for "other nine species" Table 5 provides an overview of the number of branches for the remaining nine species, B-F on the left part and G-H & K-L on the right part. First, we see the number of primary branches at all levels (I-XIV): 27 + 09 = 36 (Table 6.1),35 as a result through which Darwin solves "the riddle of the genetic code" (Section 4.3). [Review of counting through levels for primary branches is given in Tables 6.1 and 6.2.] On the right of the result, in Table 5, the result of the total number
32 .+ / 3/ H (;) OM HP8= @ / / vealed me a part of the secret. He spoke about a large outer space square and he told me that the third and the seventh number are the basis / P 33 Cf. the result 66 in the upper part and 037 in the lower part in Table 4.3 with the same pattern (66/037), also 66 in the upper part and 037 in the lower part, in Survey B.1. 34 Kedrov, 1977, p. 128, photocopy X. Having found the result where Mendeleev allegedly made a mistake in two out of three cases (!?), Kedrov concluded that even the greatest can make a mistake. In our opinion, Mendeleev did ) )' / 3 8 ' # / "#R 3 0/ B.4). 35 The results shown in Tables 6.1-6.5 refer to the "other nine species", while the results for the "all 11 species" are shown in Tables 7.1-7.5; in all of these tables, the letters on the two final branches, instead of the previous designation with small letters "a" and "z" now have the designations â and Ⱥ' # U
113 of secondary branches is given (3 + 4 = 7),36 from the zeroth to the sixth level, because there are none of them on other levels, as shown by the specific counting in the Diagram (Table 6.3).37 Therefore, the total number of branches (primary + secondary) for "other nine species," from the zeroth to the 14th level is 36 + 07 = 43 (Table 6.4), and from the zeroth to the 9th level is 32 + 07 = 39 (Table 6.5).38 In Table 6.1 we see that the number of primary branches for "other nine species", at 0-14 levels, is balanced in the odd/even positions, as well as along the two zigzag lines (18 + 18). It is clear that there is balance at levels 0-9 in odd/even positions (16 + 16), and that there is no balance for four units of the two diagonal lines (Table 6.2). For secondary branches the balance in the same spatial situations is realized with ±1 difference (3/4) (Table 6.3); for the sum of primary and secondary branches (at levels 0-14) the balance is also realized with ±1 difference (21/22) (Table 6.4), and this balance is disrupted for three units at 0-9 level (Table 6.5).
4.9. Primary and secondary branches for all 11 species Table 7.1 shows that in Darwin's diagram, we find a total of 276 branches; a number that, in union with the number 121 (which represents the total number of secondary branches of "first two species", "A" and "I"), represents the first case of a specific and unique arithmetical system (as we have shown in the Preliminaries and in Figure 4). The total number of branches splits into two sets, 60 branches in the upper part of the Diagram (with singlet branches) and 216 branches 3HR `8 # " ' # 3 :839 Table 7-2 also shows that the number of branches of the first and of the second five levels, represents a change of ±10 in relation to the arithmetic mean of the total number of branches in the lower part of the Diagram [(216:2 = 108); (108 + 10 = 118); (108-10 = 98)]. The same model is valid for the whole Diagram, for the total number of primary (Tables 7.3) and secondary branches (Table 7.4), but in relation to the total number of branches, number 276 [(276:2 = 138); (138 + 10 = 148); (138 - 10 = 128)].
36 Cf. this result 07 for the total number of secondary branches (at 0-6 level, i.e. at1-7. level), ` O' with 07 / ; U ` # O 3@@ @J@' 98 at 0-7 level, i.e. at 1-8 level. 37 # ' "#R : Z ? / 3 @ two" species "A" and "I"), but also in the logic of the secondary branches layout (in levels) for the "other nine O -' #' " ' @ species" (G-H and K-L) not any branch, neither primary nor secondary, is present at the levels after the sixth. [Notice that "nine other species" are splitting into five on the left, and four on the right.] 38 Cf. 39 all branches in "other nine species" (Tab.6.5) with all 49 primary, finalized non-fixed branches in the "first two species" ("A" and "I") (Table 1.4). 39 In the Preliminaries we have presented that here, there is also the relation between the "final" result in the genetic code (60 of "Prime Quantum 037" and 5^2 x 037) and the "final" result in Darwin's diagram (60 of "First Quantum 01" and 1 x 6^3). And the relation between the numbers 2220 and 925 in the GC is obvious (in fact it is both times determined by Pythagorean Law) while in Darwin's diagram the relation between 60 and 216 is almost unnoticeable. In the absence of a more obvious insight, we now present a possible regularity: 60 = 5 x (6 + 6) and 216 = 6 x (6 x 6).
114 Table 7.5 presents the results of the total number of branches from the zeroth to the ninth level, as in Table 7.4, of the total number of secondary branches. (A Table in analogy with Table 7.3 for the secondary branches is not possible, because there are no secondary branches in the upper part of the Diagram.) In addition to the other balances, Table 7.5 shows an obvious determination through the sequence of a series of natural numbers: 42, 43, 44, 45, 46.
4.10. Improbable and unexpected result In Section 4.3 we have shown that Darwin's equation naturally "fits" the two linear equations which determine the connection between codons and amino acids. And there is nothing surprising in that. Darwin understood (and there is no doubt about that) the existence of a specific and unique system, and with that system he adjusted his (hidden) code stored in the Diagram. However, there is another, perhaps more direct link with the genetic code, for which there is almost no explanation. / "#R ' presented in Table 4.3 to the result which represents the number of atoms in the amino acid ' # $E ' 0 )R 0 )R # = -meaning AAs is taken into account once, and in two-meaning AAs (L, S, R) twice.40 Thus, for example, for the number of nucleons in side chains of AAs he got the following result: [1 x (G1+A15+ P41+ V43+ T45 + C47 + I57+ N58 + D59 + K72 + Q72 + E73 + M75 + H81 + F91 + Y107 + W130)] + [2 x (L57 + S31 + R100)] = 1443. J' #' 0 )R ' performed with an iteration more, for the number of atoms, the result is as follows: [2 x (G1 + A4 + C5 + D7 + N8 + T8 + P8 + E10 + V10 + Q11 + M11 + H11 + I13 + F14 + Y15 +K15 + W18)] + [3 x (S5 + L13 + R17)] = 0443. On the other hand, the number of all "branch" entities/quantities in Darwin's diagram is: 276 branches (Table 4.5 in relation to Table 5) plus 46 nodes (Table 3.1) + 10 branchings (Table 3.2) equals 332. From this result, the significant differences in relation to GC are: 1443-332 = 1111 and 443-332 = 111, in both cases determined by a strict balance, expressed through the law of unity change (four and three unit positions, respectively). But that is not all. If the above iteration is derived in a Mendelevian system of AAs (Table E.1) we get the result of two parts which are related to each other also through the unit change law: 277-166 = 111. What is, however, surprising is the fact that this result written in the form 166-111-277, strict/ # "#R 066-111-177, also through the unit change law (cf. Table 4.2 with Table E.1). D # "#R diagram contains a prediction of relationships not only in terrestrial but the genetic code anywhere in the universe, under conditions of the presence of water, ammonia and methane, phosphine and hydrogen sulfide. If so, then Darwinian selection moves one step backwards in prebiotic conditions, where it refers to the choice of the life itself.
40 One-meaning AAs are decoded by the codons from one codon family, but two-meaning AAs are decoded by codons from two codon families (L,S, R).
115 4.11. More than improbable result This raises the question: whether, perhaps, it is possible to find an arithmetical system that # # "#R I ' # " ' gathered in one place? Yes, this is the system shown in the Survey B.4. Even more than that, it is / "#P / # - R hidden code (Section 4.7), as well as with the genetic code (Survey B.5 in relation to Survey B.6 and B.7), and without that unity none of these three codes [one natural (genetic code) and two created (Mendeleev code and Darwin code)] can be understood.
5. Concluding remarks
1. Presenting in this chapter "#R code, and the arguments in favor of the working hypothesis, given in the Introduction (for this and all other researches of the Diagram in future) of the actual existence of such a code, we hope that we are now also closer to # 0 )R question about the nature of arithmetical regularities in the genetic code.41 The essence of Darwin's coding is that the principle of selection must also refer to the pre-biological conditions, when it comes to selection of life itself. In some way, unknown to us, Darwin grasped and understood that biological organization must be in correspondence with the organization of unique arithmetical and/or algebraic systems; precisely as we now know that it is so in the genetic code, as presented in this, and in the previous works of several authors. Hence, the whole Darwin's book On the Origin of species is actually a qualitatively expressed biological code and the diagram represents a quantitative evidence of the same code. 2. The working hypothesis, however, can only be considered as proven, provided that one should first understand (and that is our intention, so throughout the chapter, we have provided arguments to support it) that Darwin consciously and deliberately encoded everything; in other words, it is proven that the relations presented in Darwin's diagram were not randomly presented. In addition to the aforesaid, it is enough to look at Figures 4 and 5 where two special arithmetical systems are presented, both in relation to the "arithmetical-logical square 11-12-13-14", presented in Table A.1. From the aspect of the probability theory the question is not the probability with which we can accidentally "extract" the numbers one by one, but three numbers at once [in Figure 4, the numbers are: 12-23-276, 23-34-782, etc., where the first case is Darwin's case (Table 7.1)42; in Figure 5 there are: 26-36-62, 52-60-112, etc., where the second case is
41 In one of his first works in which he presented that the physico-chemical classification of the constituents of the genetic code is followed by arithmetical patterns and the balance of the number of particles (nucleons), V. Shcherbak concluded that "The physical nature of such a phenomenon is so far not clear" (Shcherbak 1993, last sentence). 42 The number 276 as the total number of branches within Darwin's diagram. Anyway, here within the set of "possible cases" there are all two-digit, three-digit and four-digit numbers, provided that the zeroth case (1, 12, 12) is excluded; because, if it was involved, then single-digit numbers would be included as well, and the combinations would be M the combinations with repetition, so the probability would be even less.
116 Darwin's case43 (Table 2.1)]44. This, then, means that there is the question of the selection probability of not only these two arithmetical systems, but of all other arithmetical / algebraic / ' # "#R I determinants in the Diagram. 3. However, independently of the future, we present the probabilities for the two systems in Figures 4 and 5. The probability of a "favorable" event being realized, within the system in Figure 4 (for example, to "derive" the triple 12-23-276)45, the probability is 1: 6 x 10^12; and to derive all triples listed in Figure 4 (seven triples), the probability is 1: 10^79. As for the system in Figure 5, regarding the fact that the system reaches the end of the three-digit and not four-digit numbers, and that only four cases are presented, the probability is slightly higher 1: 10^33. But since these two systems are independent, with the independent events, the probability to draw both systems (in the given lengths) is 1: 10^112. It is clear that both systems in their totality, tend to reach the infinity, whereas the probability tends to reach zero, that is to say, to the impossible event. Everything would be the same if we would like to determine the appereance probability for the elements of the system, presented in the Survey B.7 (which is in a conection with the system in Survey B.6). However, in favor of the intention and the disqualification of randomness, there is a fact of conditional probability occurrence: with the appearance of the triple 177-277-377, its analogue triple 066-166-266 automatically appears; then, with the triple 288 -388-488 there is its analogue 177-277-377 etc. In addition to this, there is one fact more: the first case is additionally significant, because it contains the Darwin's solution (177-066) in the first position, and the genetic code solution (277-166) in the second position (Table E.1). 4. Based on the findings, presented in this chapter, it makes sense to set up a hypothesis (prediction!) according to which a future research will show that life, in all its levels (presented here in the unity and coherence of physical-chemical laws and arithmetical-algebraic regularities) is manifested in proportionalities and harmonious balance.46 In addition to that, we
43 The result 52+60=112 as the number of primary bran # OP OJP 3 8 /#/' within the set of "possible cases" there are all two-digit, three-digit and four-digit numbers, provided that the zeroth case (0, 12, 12) is excluded; because, if it was involved, single-digit numbers would be included as well, and the combinations would be M the combinations with repetition, so the probability would be even less. 44 Notice that arithmetical system in Figure 5 is a derivative of the system in Figure 4, of its first row. 45 Having realized that this triple is an element of another system, as well (Table C.2), which is in a strict connection with the system in Table C.1, and which is a direct determinant of the genetic code (the determinant of assignment of codons to amino acids, classified into four types of diversity), the calculation of probability practically loses its point; it becomes immediately obvious that intentions, and not coincidences are present here. At the same time, it # # / "#R gram corresponds with the structure of the genetic code, although, in the time when he lived, Darwin could not know anything about the genetic code. Simply, Darwin understood relations in arithmetical systems, presented in Tables C.1 and C.2, based on which, as we now know, the genetic code was also built. 46 O G / ' # happen in front of us during the chemical reactions of particles, have been happening up to now. A future Newton will discover the laws of these changes, as well. And, although the chemical changes are unique, they are, however, just variations on the general / # P 3- ' 465' 6698
117 expect that the results presented here will help in resolving some dilemmas - Darwinism or Intelligent design,47 as well as the dilemma: if cultural evolution is subject to Darwinian selectionism or is it a "communal exchange" (Gabora, 2013; Kaufman, 2014).48 5. It is so with hypothesis for the future, but if I am to express my opinion, here and now, just based on these results, then, here it is: Concerning the intelligent design, I have nothing to add to what I said in the previous work (here: footnote 47). As for culture, I believe that professors L. Gabora and S. Kaufman (footnote 48) are wrong. As a Darwinian selection has to move one step backwards in prebiotic conditions, it has to move one step forward, as well, where it refers to human consciousness and its "products," such as human society. All kinds of "communal exchanges" are primarily found in the input, and when it comes to the final output (which language and which culture survive and which languages and cultures disappear), they must necessarily be the result of Darwinian selection, as the most general law valid for all manifestations of life, starting with the problem of its origin in the immaterial, through all the manifestations of actual life, until the problem of appearance and manifestation of consciousness and meaningfulness, including the evolution of human society itself.
47 )64' > ?' >= OM G / 3 researches) that here, there really is a kind of intelligent design; not the original intelligent design, dealing with the question M intelligent design or evolution (Pullen, 2005), which is rightly criticized by F.S. Collins (2006). Here, ' # # O0 J " P 30HJ"8 consistent with that design which was presented by F. Castro-Chavez (2010), and is also in accordance with the Darwinism. [F. Castro-E ; 3> >' : 58= OM U ; # ; PB /' Uted that the hypothetical SPID, G' # 3 O8 0HJ"' / ' "#P ) OG 0 P 3"#' 4478' # # / ne of ) 3 )64' 449C ###) 8 AJ / / ' ) I = R spontaneous evolutionary OBP 48 L ' > 9' = OAs Gabora points out, ideas and artifacts get put to new uses and combined with one another in new ways for new functionalities, and this is what underlies technological, cultural and political evolution. None of this is captured or even approachable by way of a Darwinian theory of culture. Gabora does two things in this chapter. First, she levels a reasoned and devastating attack on the adequacy of a Darwinian theory of cultural evolution, showing that cultural evolution violates virtually all prerequisites to be encompassed by Darwin's / 0 ' # # # O
118 F I G U R E S
Figure 1.1. @ / @ "#R ) OG G 0 P 3. ' 5648
119 Figure 1.2. I / "#R (www.biologydirect/darwin)
120
Figure 1.3. "#R / ' / ` of speciesO' 59 3= L /;' Istoricheskij metod v biologii, Akademiya nauk SSSR, 1942, Moskva, Figure 15 on p. 188).
121 Figure 1.4. In mid-July 1837 Darwin started his "B" notebook on Transmutation of Species, and on page 36 he wrote "I think" above his first evolutionary tree.
122 Figure 1.5. In mid-July 1837 Darwin started his "B" notebook on Transmutation of Species, and on page 36 he wrote "I think" above his first evolutionary tree.
123 Figure 2. The logic square of the Genetic code: two single versus two double molecules; two # # # # / 3= )64' 449' 58
124 Figure 3. 0 )R -codon and non-four-codon amino acids. The one-meaning AAs are included in the sum once while two-meaning AAs (L, S, R) are included twice (Shcherbak, 1994, Fig. 1 ).
125 (0th) 01 x 12 = 012 (1) 264 (6 x 044) (1st) 12 x 23 = 276 242 (2) 506 121 (2nd) 23 x 34 = 782 242 (3) 748 (3rd) 34 x 45 = 1530 242 (4) 990 (6 x 165) (4th) 45 x 56 = 2520 242 (5) 1232 121 (5th) 56 x 67 = 3752 242 (6) 1474 (6th) 67 x 78 = 5226 242 (7) 1716 (6 x 286) (7th) 78 x 89 = 6942 242 G (50 = 49 + 01) (49 + 121 = 170) (170 + 07 = 177) (121 = 121 ± 00)
Figure 4. The multiples of [3 Z 8 3 Z 8B 3 ^ >' ' ' 8 `:7O "#R he total number of branches in the Diagram (Table 7.1); as well as the ` O # "#R / branches for two species (A and I) in the Diagram (Table 2.2).
126 0 x 13 = 00 12 012 1 x 12 = 12 50 2 x 13 = 26 10 062 3 x 12 = 36 50 4 x 13 = 52 08 112 5 x 12 = 60 50 6 x 13 = 78 06 162 7 x 12 = 84 50 8 x 13 = 104 04 212 9 x 12 = 108 (50 = 49 + 01) (49 + 121 = 170) (170 + 07 = 177)
Figure 5. The multiples of numbers 13 and 12; 13 by even, and 12 by odd numbers from natural I "#R 36 Z 7> ^ 8 ) Table 2.1.
127 01 + 00 = 01 09 + 00 = 09 02 + 02 = 04 10 + 06 = 16 03 + 01 = 04 11 + 05 = 16 01 + 00 = 01 05 + 04 = 09 04 + 00 = 04 12 + 04 = 16 02 + -01 = 01 06 + 03 = 09 GG
25 + 00 = 25 49 + 00 = 49 26 + 10 = 36 50 + 14 = 64 27 + 09 = 36 51 + 13 = 64 17 + 08 = 25 37 + 12 = 49 28 + 08 = 36 52 + 12 = 64 18 + 07 = 25 38 + 11 = 49 GG
Figure 6. The generation of the squares of natural numbers through two linear equations. "#R I I , in the area of dark tones (Tables 5 and 6.1) surrounded by two linear equations valid in the genetic code (Table C.2), presented in Survey C.2.
128 02 + 02 = 04 10 + 06 = 16 03 + 01 = 04 11 + 05 = 16 01 + 00 = 01 05 + 04 = 09 02 + 02 = 04 = 22 10 + 06 = 16 = 42 01 + 00 = 01 = 12 05 + 04 = 09 = 32 02 - 02 = 00 = 02 10 - 06 = 04 = 22 2 2 01 - 00 = 01 = 1 (?!) 05 - 04 = 01 = 1
1 - (- 1) = 2 26 + 10 = 36 50 + 14 = 64 27 + 09 = 36 51 + 13 = 64 17 + 08 = 25 37 + 12 = 49 26 + 10 = 36 = 62 50 + 14 = 64 = 82 17 + 08 = 25 = 52 37 + 12 = 49 = 72 26 - 10 = 16 = 42 50 - 14 = 36 = 62 17 - 08 = 09 = 32 37 - 12 = 25 = 52
5 - (+ 3) = 2
Figure 7. This Figure follows from the previous one, Figure 6. Three linear equations within I I R I J quadrant: two equations are valid in the genetic code (Table C.2) and one (in the middle position, dark tone) is given as Darwin's equation (Tables 5 and 6.1). + U 3"#R paradox), valid for number 1 in the first quadrant: the negative value of number 1 cannot be M negative?! [Notice also that "#R equation (27+9 = 36) is possible only in the case of squares of number 5 and 6 (25 and 36 respectively) with the changes for 01, 10 and 11, respectively, and not for any other square number pairs (cf. Tables 1.3, 1.4 and 1.5 in Chapter 2).]
129 1^2 + 2^2 + 3^2 = 14
1^1 + 2^1 + 3^1 = 06
1^3 + 2^3 + 3^3 = 6^2 1 8 27 9 + 27 = 36
G H K L 6:1 2:1 0:1 1:1 3^2 + 3^3 = 6^2 81 9 9 + 27 = 36
x^n + y^n = z^n-1 Valid only for n = 3 x^3 + y^3 = z^2 1^3 + 2^3 = 3^2 1 8 = 9
Figure 8. The relationships between the first three natural numbers. On the top area: the first row # R I I 9 M a half of the second perfect number; the second row shows the sum of the first three numbers as the first perfect number, the number 6; the third row shows that the sum of the cubes of the first three numbers equals the square of the first perfect number; in the fourth row we see the values which follow from the third ro#C # # "#R I 3 6 7 8 J ' the left there is the number of primary (bold) and secondary branches, valid for the species G, H, L' . "#R A+ hat there are two manners to "#R 5 Z ^ 4 = ` -JO position (left G & H and right K & L); and in relation to the zeroth position (there is no primary branches in K position).] In the middle area, on the right: the second variant of the generation of "#R IC ' 6 # "#= "#R I # 3 Z 5 ^ 48
130 3^2 + 2^2 + 1^2 = 14 9 4 1 5 3 3 1 1 1
2^3 + 2^2 + 2^1 =14 8 4 2
Figure 9. The relationships within the periodic system of chemical elements (PSE) in correpondence with the equation which we have taken from the first row in Figure 8; also in correpondence with the reverse form of this equation. The arrangement is as follows: 5 elements of s-type or p-type, 3 elements of d-type and 1 element of f-type. This pattern is realized (in Periodic Table) 8 times; The following pattern has 3 elements of d-type and 1 element of f- type, and it is realized 2 times; Finally, we have the form of 1 element of f-type, which is AE 5' 5> )64' 44:C / PSE, in Table 18, there are 1+14 groups ("1" as zeroth group), analogously to 1+14 elements in - R = 9 3 ' ' # known for the life of Mendeleyev, but he is still indicated it, as it is presented in Table 16, in Kedrov, 1977, p. 188); also, analogously to 1 + 14 levels in Darwin's diagram.]
131 T A B L E S
a9 03 02 z9 a9 03 02 z9 a8 03 02 z8 a8 03 02 z8 a7 05 03 z7 a7 05 20 (32) 12 03 z7 a6 05 03 z6 a6 05 03 z6 a5 04 02 z5 a5 04 02 z5
a4 04 02 z4 a4 04 02 z4 a3 04 01 z3 a3 04 01 z3 24 a2 03 02 z2 a2 03 16 ( ) 08 02 z2 a1 03 02 z1 a1 03 02 z1 a0 02 01 z0 a0 02 01 z0
Odd 19 (29)10 27 Even 17 (27) 10 29 28 / 28 (00) 36 20 56 56
Table 1.1. All primary branches at 0-9 levels (for two species: A and I) in the splitting (5 + 5). The counting starts from every initial level at which the branching occurs (0-1, 1-2, 2-?' G' 9- 10), and the 9th level is the last.
132 a10 03 02 z10 a10 03 02 z10 a9 03 02 z9 a9 03 02 z9 a8 05 03 z8 a8 05 03 z8 a7 05 03 z7 a7 05 20 (32) 12 03 z7 a6 04 02 z6 a6 04 02 z6
a5 04 02 z5 a5 04 02 z5 a4 04 01 z4 a4 04 01 z4 a3 03 02 z3 a3 03 02 z3 (24) a2 03 02 z2 a2 03 16 08 02 z2 a1 02 01 z1 a1 02 01 z1
Even 19 (29) 10 (27) 28 / 28 Odd 17 (27) 10 (29)
36 20 56 56
Table. 1.2. All primary branches at 1-10 levels (for two species: A and I) in the splitting (5+5). The counting starts from I # ; 3 ' ' ?' G' > 8' and the 10th level is the last.
133 a10 00 00 z6 a10 00 00 z10 a9 00 00 z9 a9 00 00 z9 02 ) a8 02 01 z8 a8 02 (03 01 01 z8 a7 00 00 z7 a7 00 00 z7 a6 00 00 z6 a6 00 00 z6
a5 01 00 z5 a5 01 00 z5 a4 01 00 z4 a4 01 00 z4 a3 00 01 z3 a3 00 03 (04) 01 01 z3 a2 01 00 z2 a2 01 00 z2 a1 00 00 z1 a1 00 00 z1
Even 04 (05) 01 (02) 03 / 04 Odd 01 (02) 01 (05)
05 02 07 07
Tab. 1.3. All primary, finalized, fixed branches at 1-10 levels (for two species: A and I) in the splitting (5+5). The counting is as in Table 1.2. (Notice the results in the form of the sequence: 1, 2, 3, 4, 5.)
134 a10 03 02 z10 a10 03 02 z10 a9 03 02 z9 a9 03 02 z9 a8 03 02 z8 a8 03 02 z8 a7 05 03 z7 a7 05 18 (29) 11 03 z7 a6 04 02 z6 a6 04 02 z6
a5 03 02 z5 a5 03 02 z5 a4 03 01 z4 a4 03 01 z4 a3 03 01 z3 a3 03 01 z3 (20) a2 02 02 z2 a2 02 13 07 02 z2 a1 02 01 z1 a1 02 01 z1
Even 15 (24) 09 25 25 / 24 Odd 16 (25) 09 24
18 49 49 31 (31 M 20 = 11) (29 M 18 = 11) 1 2 Fixed 7 (7 ) + 49 (7 ) non-fixed = 56 primary
Tab. 1.4. All primary, finalized, non-fixed branches at 1-10 levels (for two species: A and I) in the splitting (5+5). The counting is as in Table 1.2.
135 a14 08 06 z14 a14 08 06 z14 a13 08 06 z13 a13 08 06 z13 32 56 a12 08 06 z12 a12 08 ( ) 24 06 z12 a11 08 06 z11 a11 08 06 z11 a10 00 00 z10 a10 00 00 z10
a10 03 02 z10 a10 03 02 z10 a9 03 02 z9 a9 03 02 z9 29) a8 03 02 z8 a8 03 18 ( 11 02 z8
a7 05 03 z7 a7 05 03 z7 a6 04 02 z6 a6 04 02 z6
a5 03 02 z5 a5 03 02 z5 a4 03 01 z4 a4 03 01 z4 a3 03 01 z3 a3 03 13 (20) 07 01 z3 a2 02 02 z2 a2 02 02 z2 a1 02 01 z1 a1 02 01 z1
Even 31 (52) 21 53 49 / 56 Odd 32 (53) 21 52 63 42 105 (216 M 111) (105 = 56 + 49) [233-105 = 128 (121+7)]
Tab. 1.5. All primary, finalized, non-fixed branches on 1-14 levels (for two species: A and I) in the splitting (3x5). The counting is as in Table 1.2. Notice the self-similarity expressed through quantities on two zigzag lines: 49 as non-fixed branches (Table 1.4), 56 as total number of primary branches in the lower as well as in the upper part of the Diagram (Table 2.1). The result 105 follows from this distinction: all 112 primary branches (Table 2.1) minus 7 fixed branches (Table 1.3). The balance and self-similarity: 105 as all primary, finalized, non-fixed branches = 216 as all the branches in the lower part of the Diagram (0-9 levels, for all 11 species) minus 111 O P A0-similarity is present here because 111M105 = 6 and 177 (in Table 4.1) minus 111 equals 66 as in Table 4.2 (Notice the determinants 6 and 66 in Table B.1).]
136 a14 08 06 z14 a14 08 06 z14 a13 08 06 z13 a13 08 06 z13 56 a12 08 06 z12 a12 08 32 ( ) 24 06 z12 a11 08 06 z11 a11 08 06 z11 a10 00 00 z10 a10 00 00 z10
a9 03 02 z9 a9 03 02 z9 a8 03 02 z8 a8 03 02 z8 a7 05 03 z7 a7 05 20 (32) 12 03 z7 a6 05 03 z6 a6 05 03 z6 a5 04 02 z5 a5 04 02 z5
a4 04 02 z4 a4 04 02 z4 a3 04 01 z3 a3 04 01 z3 24 a2 03 02 z2 a2 03 16 ( ) 08 02 z2 a1 03 02 z1 a1 03 02 z1 a0 02 01 z0 a0 02 01 z0
Even 33 (55) 22 57 52 / 60 Odd 35 (57) 22 55
08 06 14 (well-marked on 14th level) 24 18 42 (non-marked on 11-13th levels) 32 24 56 36 20 56 (well-marked on 00-09th levels) 68 44 112 (4 x 28) total 44 26 70 well-marked (with letters)
Tab. 2.1. All primary branches for two species, "A" and "I", with the splitting into (3 x 5) levels. The pattern 52+62 = 112 appears to be the middle case in a specific arithmetical system (Figure 5). Notice that 56 branches are in the upper as well as in the lower part of the Diagram. (Notice the differences between pattern 52 / 60 / 112, valid for all primary branches (in this Table) and the pattern 62 / 59 / 121 (in Table 2.2), valid for secondary branches, where the changes are ±10 and ±01. Notice also that the first pattern 52 / 60 / 112 is the middle case within a specific arithmetical system, presented in Figure 5.)
137 a9 08 05 z9 a9 08 05 z9 a8 09 06 z8 a8 09 06 z8 a7 13 07 z7 a7 13 46 (74) 28 07 z7 a6 06 04 z6 a6 06 04 z6 a5 10 06 z5 a5 10 06 z5
a4 09 04 z4 a4 09 04 z4 a3 07 03 z3 a3 07 03 z3 a2 05 04 z2 a2 05 31 (47) 16 04 z2 a1 06 01 z1 a1 06 01 z1 a0 04 04 z0 a0 04 04 z0
Odd 44 ( 66) 22 (55) 62 / 59 Even 33 (55) 22 (66) 77 (11) 44 121 (56 + 65) Middle pair 60/61 vs 62/59 as result
Table 2.2. All secondary branches for two species, "A" and "I", with the splitting into (5 + 5) levels. There are none of them after the 9th level. [Cf. pattern 74/77 with the pattern 64/66 in Table 4.1; then 44/46 with the pattern 64/66 also in Table 4.1.]
138 a9 03 02 z9 a9 03 02 z9 a8 03 02 z8 a8 03 02 z8 a7 05 03 z7 a7 05 18 (29) 11 03 z7 a6 04 02 z6 a6 04 02 z6 a5 03 02 z5 a5 03 02 z5
a4 03 01 z4 a4 03 01 z4 a3 03 01 z3 a3 03 01 z3 (17) a2 02 02 z2 a2 02 11 06 02 z2 a1 02 01 z1 a1 02 01 z1 a0 01 01 z0 a0 01 01 z0
Odd 16 (25) 09 (22) 24 / 22 Even 13 (21) 08 (24) 29 17 46 46 46 + 10 = 56
Tab. 3.1. All nodes for two species, "A" and "I", with the splitting into (5+5) levels. The balances are self-evident. [Notice a special balance: 46 nodes + 10 branchings (Tables 3.1 and 3.2) equals 56 group tree-entities in correspondence with 56 primary branches (Table 1.1) as individual tree-entities.]
139 a6 1 1 z6 a6 1 1 z6 a5 1 0 z5 a5 1 03 (05) 02 0 z5 a4 1 1 z4 a4 1 1 z4
a3 1 0 z3 a3 1 0 z3 a2 1 0 z2 a2 1 0 z2 a1 1 1 z1 a1 1 04 (05) 01 1 z1 a0 1 0 z0 a0 1 0 z0
Even 04 ( 06) 02 (05) 04 /06 Odd 03 (04) 01 (05)
07 03 10 10 10 + 40 = 56 (cf. legend in Tab. 3.1)
Tab. 3.2. All branchings for two species, "A" and "I", with the splitting into (4+3) levels. This is due to the fact that there are branchings in the Diagram just from the zeroth to the 6th level. This finding requires that in the analysis of the number of all branches, except for splitting into the (5+5) levels as in Table 4.1, we must as well analyze the splitting into (7+3) levels as in Table 4.2, and then into (3+4+3) as in Table 4.3 and (3+2+2+3) as in Table 4.4. The balances are self- evident. [Notice that the left tree of the Diagram (Figure 1.1) contains two large branches; and on the left branch there are only two branchings (bold, underlined units in the second column).]
140 a6 1 1 z6 a6 1 1 z6 a5 1 0 z5 a5 1 0 z5 (06) a4 1 1 z4 a4 1 04 02 1 z4 a3 1 0 z3 a3 1 0 z3
a2 1 0 z2 a2 1 0 z2 a1 1 1 z1 a1 1 03 (04) 01 1 z1 a0 1 0 z0 a0 1 0 z0
Even 04 (06) 02 (05) 05 /05 Odd 03 (04) 01 (05) 07 03 10 10 10 + 40 = 56 (cf. legend in Tab. 3.1)
Tab. 3.3. All branchings for two species, "A" and "I", with the splitting into (3+4) levels as a reverse way in relation to Table 3.2. Notice that the splitting of 7 levels into 3 and 4 (3+4=7) represent a correspondence with the Lucas numbers series at the same time (Figure D.1).
141 a9 11 07 z9 a9 11 07 z9 a8 12 08 z8 a8 12 08 z8 a7 18 10 z7 a7 18 66 (106) 40 10 z7 a6 11 07 z6 a6 11 07 z6 a5 14 08 z5 a5 14 08 z5
a4 13 06 z4 a4 13 06 z4 a3 11 04 z3 a3 11 04 z3 a2 08 06 z2 a2 08 47 (71) 24 06 z2 a1 09 03 z1 a1 09 03 z1 a0 06 05 z0 a0 06 05 z0
Odd 63 (95) 32 (82) 90 / 87 Even 50 (82) 32 (95)
113 64 177 177 (177 = 88+89) (90-89 = 01) (88-87 = 01)
Tab. 4.1. All branches (primary + secondary) for two species, "A" and "I", with the splitting into (5+5) levels. The pattern 90/87 appears to be an inverse result 80/97 which appears by the splitting into (7+3) levels (Table 4.2) and a strict balance in relation to 89/88 (the balance in frame of ±1) by the splitting into (3+4+3) levels (Table 4.3). [Cf. pattern 64/66 with pattern 74/77 and pattern 44/46 in Table 2.2.]
142 a9 11 07 z9 a9 11 07 z9 66) a8 12 08 z8 a8 12 41 ( 25 08 z8 a7 18 10 z7 a7 18 10 z7
a6 11 07 z6 a6 11 07 z6 a5 14 08 z5 a5 14 08 z5 a4 13 06 z4 a4 13 06 z4 a3 11 04 z3 a3 11 72 (111) 39 04 z3 a2 08 06 z2 a2 08 06 z2 a1 09 03 z1 a1 09 03 z1 a0 06 05 z0 a0 06 05 z0
Odd 63 ( 95) 32 82 80 / 97 Even 50 (82) 32 95 113 64 177 177 (066-111-177) vs (166-111-277) in Tab. E.1
Tab. 4.2. All branches (primary + secondary) for two species, "A" and "I", with the splitting into (7+3) levels with pattern 80/97 corresponding to the pattern 90/87 which appears by the splitting into (5+5) levels in Table 4.1. On the other hand pattern 066-111-177 corresponds to pattern 166-111-277 in genetic code (Appendix E). All other balances are self-evident.
143 a9 11 07 z9 a9 11 07 z9 (66) a8 12 08 z8 a8 12 41 25 08 z8 a7 18 10 z7 a7 18 10 z7
a6 11 07 z6 a6 11 07 z6 a5 14 08 z5 a5 14 49 (74) 25 08 z5 a4 z4 a4 z4 13 06 13 06 a3 11 04 z3 a3 11 04 z3
a2 08 06 z2 a2 08 06 z2 a1 09 03 z1 a1 09 23 (37) 14 03 z1 a0 06 05 z0 a0 06 05 z0
Odd 63 ( 95) 32 82 89 / 88 Even 50 (82) 32 95 113 64 177 177 37 + 74 = 111
Tab. 4.3. This Table follows from Table 4.2. The formal splitting into (3+4+3) levels corresponds to an extended Cantor triadic set (Figure D.2). On the other hand, the number of the branches # 0 )R I O / P 3 8 # I O / / P 3>?: Z >:98 # I >?: OH I >?:PC I number 66, and altogether in connection with a specific and unique arithmetical system (Table B.1 and Survey B.1 in Appendix B).
144 a9 11 07 z9 a9 11 07 z9 (66) a8 12 08 z8 a8 12 41 25 08 z8 a7 18 10 z7 a7 18 10 z7
(40) a6 11 07 z6 a6 11 25 15 07 z6 a5 14 08 z5 a5 14 08 z5
a4 13 06 z4 a4 13 06 z4 (34) a3 11 04 z3 a3 11 24 10 04 z3
a2 08 06 z2 a2 08 06 z2 a1 09 03 z1 a1 09 23 (37) 14 03 z1 a0 06 05 z0 a0 06 05 z0
63 ( 95) 32 82 94 / 83 50 (82) 32 95 113 (13) 64 177 177 (94/83 vs 82/95) (94-83 = 11)
Tab. 4.4. All branches (primary + secondary) for two species, "A" and "I", with the splitting into (3+2+2+3) levels. The balances are self-evident.
145 a14 08 06 z14 a14 08 06 z14 a13 08 06 z13 a13 08 06 z13 56 a12 08 06 z12 a12 08 32 ( ) 24 06 z12 a11 08 06 z11 a11 08 06 z11 a10 00 00 z10 a10 00 00 z10
a9 11 07 z9 a9 11 07 z9 a8 12 08 z8 a8 12 08 z8 a7 18 10 z7 a7 18 66 (106) 40 10 z7 a6 11 07 z6 a6 11 07 z6 a5 14 08 z5 a5 14 08 z5
a4 13 06 z4 a4 13 06 z4 a3 11 04 z3 a3 11 04 z3 a2 08 06 z2 a2 08 47 (71) 24 06 z2 a1 09 03 z1 a1 09 03 z1 a0 06 05 z0 a0 06 05 z0
Even 66 (110) 44 123 114 / 119 Odd 79 (123) 44 110 145 88 233 233 (233 = 116 + 117)
Tab. 4.5. All branches (primary + secondary) for two species, "A" and "I", with the splitting into (3 x 5) levels, 0-14. The balances are self-evident.
146 Primary Secondary B 00 06 G B 01 01 G C 01 02 H C 01 01 H D 02 00 K D 01 01 K E 10 01 L E 00 01 L F 14 F 00
27 09 03 04 36 (43) 07
(233 + 43 = 276) (276 + 56 = 332) 99
up 276 = 216down + 60
Table 5. All branches (primary + secondary) for "other nine species" for the left and the right " ' 6 I : Z >4 ^ ?7 "#R equation, valid to determination of the genetic code (Figure 6, 7 & 8 and Table 6.1); and the equation 03 + 04 = 07 corresponds to the first three members of Lucas number series (Figure D.1).The number 233 comes from Table 4.5 and together with this result (43) makes 276 which is the total number of branches within the Diagram. In addition: 56 = 46 nodes plus 10 ' O P FI I ?? 233.
147 â14 01 00 Ⱥ14 â14 01 00 Ⱥ14 â13 01 00 Ⱥ13 â13 01 00 Ⱥ13 â12 01 00 Ⱥ12 â12 01 04 (04) 00 00 Ⱥ12 â11 01 00 Ⱥ11 â11 01 00 Ⱥ11 â10 00 00 Ⱥ10 â10 00 00 Ⱥ10
â9 02 00 Ⱥ9 â9 02 00 Ⱥ9 â8 02 00 Ⱥ8 â8 02 00 Ⱥ8 â7 02 00 Ⱥ7 â7 02 10 (11) 01 00 Ⱥ7 â6 02 00 Ⱥ6 â6 02 00 Ⱥ6 â5 02 01 Ⱥ5 â5 02 01 Ⱥ5
â4 02 01 Ⱥ4 â4 02 01 Ⱥ4 â3 02 01 Ⱥ3 â3 02 01 Ⱥ3 â2 02 01 Ⱥ2 â2 02 13 (21) 08 01 Ⱥ2 â1 03 02 Ⱥ1 â1 03 02 Ⱥ1 â0 04 03 Ⱥ0 â0 04 03 Ⱥ0
Odd 13 (17) 04 (18) 18 / 18 Even 14 (19) 05 (18)
01 00 01 (well-marked on 14th level) 03 00 03 (non-marked on 11-13th levels) 04 00 04 02 00 02 (well-marked on 0-9th levels)
09 30 (non-marked on 0-9th levels) 21 24 09 33 (non-marked on 0-14th levels) 09 36 total 27 (18 = 28 M 10) (112 + 36 = 148)
Table 6.1. All primary branches for 9 species (B, C, D, E, F on the left and G, H, K, L on the right) at 0- 9 "#R I 3: Z >4 ^ ?78 3 D 7 < :8
148 â9 02 00 Ⱥ9 â9 02 00 Ⱥ9 â8 02 00 Ⱥ8 â8 02 00 Ⱥ8 â7 02 00 Ⱥ7 â7 02 10 (11) 01 00 Ⱥ7 â6 02 00 Ⱥ6 â6 02 00 Ⱥ6 â5 02 01 Ⱥ5 â5 02 01 Ⱥ5
â4 02 01 Ⱥ4 â4 02 01 Ⱥ4 â3 02 01 Ⱥ3 â3 02 01 Ⱥ3 21 â2 02 01 Ⱥ2 â2 02 13 ( ) 08 01 Ⱥ2 â1 03 02 Ⱥ1 â1 03 02 Ⱥ1 â0 04 03 Ⱥ0 â0 04 03 Ⱥ0 Odd 11 (15) 04 (16) 18 / 14 Even 12 (17) 05 (16) 23 09 32 32 56 + 32 = 88
Table 6.2. Primary branches for 9 species (B, C, D, E, F on the left and G, H, K, L on the right) at 0-9 levels.
149 â9 00 00 Ⱥ9 â9 00 00 Ⱥ9 â8 00 00 Ⱥ8 â8 00 00 Ⱥ8 â7 00 00 Ⱥ7 â7 00 00 (01) 01 00 Ⱥ7 â6 00 01 Ⱥ6 â6 00 01 Ⱥ6 â5 00 00 Ⱥ5 â5 00 00 Ⱥ5
â4 00 00 Ⱥ4 â4 00 00 Ⱥ4 â3 00 00 Ⱥ3 â3 00 00 Ⱥ3 03 â2 01 01 Ⱥ2 â2 01 (06) 03 01 Ⱥ2 â1 01 01 Ⱥ1 â1 01 01 Ⱥ1 â0 01 01 Ⱥ0 â0 01 01 Ⱥ0
Odd 01 ( 02) 01 03 03 / 04 Even 02 (05) 03 04 03 04 07 07 121 + 07 = 128
Table 6.3. All secondary branches for 9 species (B, C, D, E, F on the left and G, H, K, L on the right) at 0-9 levels.
150 â14 01 00 Ⱥ14 â14 01 00 Ⱥ14 â13 01 00 Ⱥ13 â13 01 00 Ⱥ13 â12 01 00 Ⱥ12 â12 01 04 (04) 00 00 Ⱥ12 â11 01 00 Ⱥ11 â11 01 00 Ⱥ11 â10 00 00 Ⱥ10 â10 00 00 Ⱥ10
â9 02 00 Ⱥ9 â9 02 00 Ⱥ9 â8 02 00 Ⱥ8 â8 02 00 Ⱥ8 â7 02 00 Ⱥ7 â7 02 10 (12) 02 00 Ⱥ7 â6 02 01 Ⱥ6 â6 02 01 Ⱥ6 â5 02 01 Ⱥ5 â5 02 01 Ⱥ5
â4 02 01 Ⱥ4 â4 02 01 Ⱥ4 â3 02 01 Ⱥ3 â3 02 01 Ⱥ3 27 â2 03 02 Ⱥ2 â2 03 16 ( ) 11 02 Ⱥ2 â1 04 03 Ⱥ1 â1 04 03 Ⱥ1 â0 05 04 Ⱥ0 â0 05 04 Ⱥ0
Even 16 (24) 08 (22) Odd 21 / 22 14 (19) 05 (21)
30 13 43 43
Table 6.4. All branches (primary + secondary) for 9 species (B, C, D, E, F on the left and G, H, K, L on the right) at 0-14 levels. Notice the balances: 21/22 versus 19/24 as a change for ±2; then: 27 as 9 x 3 and 30 as 10 x 3.
151 â9 02 00 Ⱥ9 â9 02 00 Ⱥ9 â8 02 00 Ⱥ8 â8 02 00 Ⱥ8 â7 02 00 Ⱥ7 â7 02 10 (12) 02 00 Ⱥ7 â6 02 01 Ⱥ6 â6 02 01 Ⱥ6 â5 02 01 Ⱥ5 â5 02 01 Ⱥ5
â4 02 01 Ⱥ4 â4 02 01 Ⱥ4 â3 02 01 Ⱥ3 â3 02 01 Ⱥ3 27 â2 03 02 Ⱥ2 â2 03 16 ( ) 11 02 Ⱥ2 â1 04 03 Ⱥ1 â1 04 03 Ⱥ1 â0 05 04 Ⱥ0 â0 05 04 Ⱥ0 Odd 12 (17) 05 (19) 21 / 18 Even 14 (22) 08 (20) 26 13 39 39 The sums: 17, 18, 19, 20, 21, 22
Table 6.5. All branches (primary + secondary) for 9 species (B, C, D, E, F on the left and G, H, K, L on the right) at 0-9 levels.
152 â14 09 06 Ⱥ14 â14 09 06 Ⱥ14 â13 09 06 Ⱥ13 â13 09 06 Ⱥ13 60 â12 09 06 Ⱥ12 â12 09 36 ( ) 24 06 Ⱥ12 â11 09 06 Ⱥ11 â11 09 06 Ⱥ11 â10 00 00 Ⱥ10 â10 00 00 Ⱥ10
â9 13 07 Ⱥ9 â9 13 07 Ⱥ9 â8 14 08 Ⱥ8 â8 14 08 Ⱥ8 â7 Ⱥ7 â7 76 (118) Ⱥ7 20 10 20 42 10 â6 13 08 Ⱥ6 â6 13 08 Ⱥ6 â5 16 09 Ⱥ5 â5 16 09 Ⱥ5
â4 15 07 Ⱥ4 â4 15 07 Ⱥ4 â3 13 05 Ⱥ3 â3 13 05 Ⱥ3 98 â2 11 08 Ⱥ2 â2 11 63 ( ) 35 08 Ⱥ2 â1 13 06 Ⱥ1 â1 13 06 Ⱥ1 â0 11 09 Ⱥ0 â0 11 09 Ⱥ0
Even 82 (134) 52 145 141 / 135 Odd 93 (142) 49 131 175 101 276 276 (1 x 496) M 220 = 276) (496 M 284 = 112 + 100) (2 x 028) + 220 = 276
Table 7.1. All branches (primary + secondary) for all the 11 species at 0-14 levels. Notice the balances: 131/145 versus 141/135 as a change for ±10; then 141/135 versus 142/134 as a change for ±1. Notice also the relations to the second (28) and the third (496) perfect number as well as the relation to the first pair of friendly numbers (220 and 284). In addition: the total number of branches (276) appears to be the first case in a specific and unique arithmetical system (Figure 4).
153
â9 13 07 Ⱥ9 â9 13 07 Ⱥ9 â8 14 08 Ⱥ8 â8 14 08 Ⱥ8 â7 20 10 Ⱥ7 â7 20 76 (118) 42 10 Ⱥ7 â6 13 08 Ⱥ6 â6 13 08 Ⱥ6 â5 16 09 Ⱥ5 â5 16 09 Ⱥ5
â4 15 07 Ⱥ4 â4 15 07 Ⱥ4 â3 13 05 Ⱥ3 â3 13 05 Ⱥ3 98 â2 11 08 Ⱥ2 â2 11 63 ( ) 35 08 Ⱥ2 â1 13 06 Ⱥ1 â1 13 06 Ⱥ1 â0 11 09 Ⱥ0 â0 11 09 Ⱥ0 Odd 75 (112) 37 (101) 111 / 105 Even 64 (104) 40 (115) 139 77 216 216
Table 7.2. All branches (primary + secondary) for all the 11 species at 0M9 levels. Notice the balances: 101/115 versus 111/105 as a change for ±10; then 111/105 versus 112/104 as a change / + 7 HR ' /' 7 (3^3 + 4^3 + 5^3 = 6^3 = 216). The results 98/108 appear to be in relation to a half HR number, as a change for ±10 (108 ±10).
154 â14 09 06 Ⱥ14 â14 09 06 Ⱥ14 â13 09 06 Ⱥ13 â13 09 06 Ⱥ13 â12 09 06 Ⱥ12 â12 09 36 (60) 24 06 Ⱥ12 â11 09 06 Ⱥ11 â11 09 06 Ⱥ11 â10 00 00 Ⱥ10 â10 00 00 Ⱥ10
â9 05 02 Ⱥ9 â9 05 02 Ⱥ9 â8 05 02 Ⱥ8 â8 05 02 Ⱥ8 â7 07 03 Ⱥ7 â7 07 30 (43) 13 03 Ⱥ7 â6 07 03 Ⱥ6 â6 07 03 Ⱥ6 â5 06 03 Ⱥ5 â5 06 03 Ⱥ5
â4 06 03 Ⱥ4 â4 06 03 Ⱥ4 â3 06 02 Ⱥ3 â3 06 02 Ⱥ3 45 â2 05 03 Ⱥ2 â2 05 29 ( ) 16 03 Ⱥ2 â1 06 04 Ⱥ1 â1 06 04 Ⱥ1 â0 06 04 Ⱥ0 â0 06 04 Ⱥ0
Even 47 (74) 27 75 78 / 70 Odd 48 (74) 26 73 95 53 148 148
Table 7.3. All primary branches for all the 11 species at 0M14 levels. The total number 148 appears to be in relation to the half of the total number of branches (of number 276 from Table 7.1) (148 = 138 + 10). Notice the balances: 78/70 in this Table versus 68/60 in Table 7.4 as a change for ±10; then 74/74 versus 73/75 as a change for ±1. The result 43/45 appears to be in relation to the arithmetic mean 44/44 as a change for ±1.
155 â9 08 05 Ⱥ9 â9 08 05 Ⱥ9 â8 09 06 Ⱥ8 â8 09 06 Ⱥ8 â7 13 07 Ⱥ7 â7 13 46 (75) 29 07 Ⱥ7 â6 06 05 Ⱥ6 â6 06 05 Ⱥ6 â5 10 06 Ⱥ5 â5 10 06 Ⱥ5
â4 09 04 Ⱥ4 â4 09 04 Ⱥ4 â3 07 03 Ⱥ3 â3 07 03 Ⱥ3 â2 06 05 Ⱥ2 â2 06 34 (53) 19 05 Ⱥ2 â1 07 02 Ⱥ1 â1 07 02 Ⱥ1 â0 05 05 Ⱥ0 â0 05 05 Ⱥ0
Odd 45 ( 70) 25 60 65 / 63 Even 35 (58) 23 68 80 48 128 128 121 + 7 = 128
Table 7.4. All secondary branches for all the 11 species at 0M9 levels. [The secondary branches do not exist in the upper part of the Diagram (levels 11-14)]. The total number 128 appears to be in relation to the half of the total number of branches (of number 276 from Table 7.1) (128 = 138 - 10). Notice the balances: 60/68 versus 70/58 as a change for ±10; then 68/60 in this Table versus 78/70 in Table 7.3 as a change for ±10; then 74/74 in Table 7.3 versus 64 ±1 in this Table.
156 â9 05 02 Ⱥ9 â9 05 02 Ⱥ9 â8 05 02 Ⱥ8 â8 05 02 Ⱥ8 â7 07 03 Ⱥ7 â7 07 30 (43) 13 03 Ⱥ7 â6 07 03 Ⱥ6 â6 07 03 Ⱥ6 â5 06 03 Ⱥ5 â5 06 03 Ⱥ5
â4 06 03 Ⱥ4 â4 06 03 Ⱥ4 â3 06 02 Ⱥ3 â3 06 02 Ⱥ3 45 â2 05 03 Ⱥ2 â2 05 29 ( ) 16 03 Ⱥ2 â1 06 04 Ⱥ1 â1 06 04 Ⱥ1 â0 06 04 Ⱥ0 â0 06 04 Ⱥ0
Even 29 (44) 15 (45) 46 / 42 Odd 30 (44) 14 (43)
59 (88) 29 88 88 56 + 32 = 88) (59 + 36 = 95)
Table 7.5. All primary branches for all the 11 species at 0M9 levels. The total number 88 as a result of 148 (all primary branches in Table 7.3) minus 60 branches in the upper part of the Diagram at levels 11-14 (Table 7.3). Notice the balances: 44/44 versus 43/45 as a change for ±1; then 43/45 versus 42/46 as a change for ±1; then 29/30 in even/odd positions versus 29/30 in up/down positions; also15/14 in even/odd positions versus 16/13 in up/down positions.
157 Appendix A
Table A.1. The arithmetical logic square: the space of the maximum possible inversions within decimal numbering system (Rako64' 449' ?68
158 Appendix B
5 F 14 15 Y 4 L 13 04 A 3 Q 11 08 N 2 P 08 13 I 1 T 08 11 M 1 S 05 05 C 2 G 01 10 V 3 D 07 10 E 4 K 15 17 R 5 H 11 18 W
O E/ J H 0/ 3EJH08 G J position there are chalcogene AAs (S, T & C, M); then M U `/O M there are the AAs of non-alaninic stereochemical types (G, P & V, I), then two double acidic AAs with their two amide derivatives (D, E & N, Q), the two original aliphatic AAs with two amine derivatives (A, L & K, R); and, finally, four aromatic AAs (F,Y & H, W) M two up and two down. The said five classes belong to two super classes: primary superclass in light areas and secondary superclass in dark areas. Notice that each amino acid position in this CIPS is strictly determined and none of P 3 )64' >>4' ?C > ' D 8
159 D E D E K R 6 x 10 = 60 60 K R 60 H W 6 x 09 = 54 H W L A 6 x 09 = 54 L A 6 x 06 = 36 54 54 Q N Q N P I P I 54 M 10 = 44 G V 36 + 10 = 46 54 T M 46 F Y F Y T M S C 44 S C 36 G V
(6 x 1) x 10 = 60 + (6 x 0) = 6 x 10 = 60 G S T P Q L F O P [(6 x 1) x 10 = 60] + (6 x 0) + (6 x 1) = 6 x 11 = 66 V C M I N A Y (their complements) [(6 x 1) x 10 = 60] + (6 x 0) + (6 x 1) + (6 x 2) = 6 x 13 = 78 D K H / E R W (their non-complements)
Figure B.2. This Figure follows from CIPS, presented in Figure B.1. First, there are five charged AAs. Then three other quintets follow in accordance to the three principles: principle of minimum change, principle of continuity and principle of dense packing. As it is self-evident, the system is determined by the first perfect number M the number 6. For the lower part of the Figure cf. the determination of GC by Golden mean (Rako64' 445a).
160 Multiples of 01, 6, 66, 666, 037 01 G 6 G 66 G 666 G 037 162 = 216 M (2 x 27) 27 162 1782 162 999 26 156 1716 17316 962 25 150 1650 16650 925 G 13 78 858 8658 481 12 72 792 7992 444 11 66 726 7326 407 G 03 18 198 1998 111 02 12 132 1332 074 01 6 66 666 037 7 HR 36^3 = 216)
Table B.1. The multiples of the numbers are presented in the first row. The 13th case is the sum of the first four perfect numbers (6 + 28 + 496 + 8128 = 8658).
7 ^ F? ^ 3>??? G8 U 5 6 x 11 = 66 (60 + 06) 77 ^ F? ^ > 777 G8 U 5 66 x 11 = 726 (660 + 066) 666 = 111/3 = 037 x 18 666 x 11 = 7326 (6660 + 0666)
(1 x 037) + (2 x 037) = 111 111 + 66 = 177
Survey B.1. O P 3778 3>?:8 "#R 9?
161 Multiples of 01, 7, 77, 777, 037 01 G 7 G 77 G 777 G 037 189 = 216 M (1 x 27) 27 189 2079 20979 999 26 182 2002 20202 962 25 175 1925 19425 925 G 13 91 1001 10101 481 12 84 924 9324 444 11 77 847 8547 407 G 03 21 231 2331 111 02 14 154 1554 074 01 7 77 777 037 7 HR 36^3 = 216)
Table B.2. The multiples of the numbers presented in the first row. The 13th case corresponds to U 3 # 8 / 3 )64' 1998).
7 = 1/3 = 3>??? G8 U 7 x 11 = 77 (70 + 07) :: ^ F? ^ > 777 G8 U 77 x 11 = 847 (770 + 077) 777 = 111/3 = 037 x 21 777 x 11 = 8547 (7770 + 0777)
(1 x 037) + (2 x 037) = 111 111 + 77 = 188
Survey B.2. O P 3::8 and the first integer case (037) "#R 0/ (9 3 # ) 8
162 Multiples of 01, 8, 88, 888, 037 01 G 8 G 88 G 888 G 037 216 = 216 ± (0 x 27) 27 216 2376 23976 999 26 208 2288 23088 962 25 200 2200 22200 925 G 13 104 1144 11544 481 12 96 1056 10656 444 11 88 968 9768 407 G 03 24 264 2664 111 02 16 176 1776 074 01 8 88 888 037 (3^3 = 27) (6^3 = 216)
Table B.3. # HR 7 3 78 O5P
5 ^ F? ^ 3>??? G8 U 9 8 x 11 = 88 (80 + 08) 55 ^ F? ^ > 777 G8 U 9 88 x 11 = 968 (880 + 088) 888 = 111/3 = 037 x 24 888 x 11 = 9768 (8880 + 0888)
(1 x 037) + (2 x 037) = 111 111 + 88 = 199
Survey B.3. O P 3558 3>?:8 "#R s in Survey B.4 (middle area in dark tones).
163 (1 x 037) + (2 x 037) = 111 111 + 66 = 177 177 M 65 = 112 27 x 037 = 999 177 M 56 = 121 121+112 = 233 (30 / 37 / 77) 6^1 = 6 177 + 077 = 254 254 = 117 +137 (30 / 27 / 67) 5^2 = 25 (31) 177 = 50+127 254 = 50 + 204
(1 x 037) + (2 x 037) = 111 111 + 77 = 188 188 M 76 = 112 27 x 037 = 999 188 M 67 = 121 121+112 = 233 (30 / 37 / 77) 7^1 = 7 188 + 088 = 276 276 = 128 +148 (30 / 27 / 67) 6^2 = 36 (43) 188 = 60+128 276 = 60 + 216
(1 x 037) + (2 x 037) = 111 111 + 88 = 199 199 M 87 = 112 27 x 037 = 999 199 M 78 = 121 121+112 = 233 (30 / 37 / 77) 8^1 = 8 199 + 099 = 298 298 = 139 +159 (30 / 27 / 67) 7^2 = 49 (57) 199 = 70+129 298 = 70 + 228
Survey B.4. The first area corresponds to Table B.1 and Survey B.1; the second (in dark tones) to Table B.2 and Survey B.2; and the third area corresponds to Table B.3 and Survey B.3. The middle area is especially significant because it, mutatis mutandis' "#R I - R quantitatives (the same area, on the left: 30/37/77 versus ?>F:F7:8 3 0 9:' ' - R / M Photocopy X in Kedrov, 1977, pp. 128-129).
164 IV V VI IV V VI VII 6 7 8 C N O (3)
12 (14) 16 H C N O (4) 12 13 15 16 U C A G 34 35 37 38 H C N O (5) (36) (P) S
Survey B.5. A hypothetical model for the connection between the quantities/entities in Tables B.1, B.2 and B.3 and 6-7-8 proton determined chemical elements (C-N-O) as constituents of life anywhere in the universe. On the left: 6, 7, 8 protons for first three elements in IV-V-VI group of Periodic system of chemical elements, respectively; then 12, 14, 16 nucleons of these elements; then 12, 13, 15, 16 atoms in four Py/Pu bases, with the relation to the half of second perfect number (28); in the last row, there is the number of atoms within four nucleotide molecules in relation to the cube of the first perfect number, number 6. [Notice that the number of nucleons in the second row and the number of atoms in the third row represent a unique type of self-similarity.] On the right: 3, 4 and 5 chemical elements as constituents of protein amino acids M the constituents of proteins. Notice that the last case on the right represents five elements in amino acid molecules (C,N,O,S,H) and five elements in nucleotide molecules (C,N,O,P,H) at the same time. Notice also that hydrogen, as a nonmetal, exists within the seventh group of Periodic system. Altogether it is self-evident that the neighbor positions of life-elements are determined with the three principles: principle of minimum change, principle of continuity and the principle of neighborhood.
165 (6) 1332 (6) 832 2553 1553 (5) 1221 (5) 721 2331 1331 (4) 1110 (4) 610 2109 1109 (3) 999 (3) 499 1887 887 (2) 888 (2) 388 1665 665 (1) 777 6A616 (1) 277 [6(10)6]16 (111) 1443 ½ [1660]10 (111) 443 (1) 666 (1) 166 1221 221 (2) 555 (2) 055 999 -001 (3) 444 (3) -056 777 -223 (4) 333 (4) -167 555 -445 (5) 222 (5) -278 333 -667 (6) 111 (6) -389
Survey B.6. If multiples 666 (Table B.1) and 777 (Table B.2) have a middle position within the system of presented multiples, then it becomes obvious that there are the relations to the number of nucleons as well as of atoms within amino acid molecules as constituents of the Genetic code. Number 1443 as the number of nucleons within 23 amino acid molecules, within their side ' 0 )R 3D ?8 + 99? F7 four perfect numbers (6+28+496+8128 = 8658 = 6 x 1443) and the sum of all multiples in the second column of this Table M ? O P 3 functional groups) there are 1702 nucleons written in decimal numbering system, or 6(10)6 (i.e. 6A6) in hexadecimal system (see the window in the middle frame area). Number 443 as the number of atoms within 43 amino acid molecules (within their side chains) after the arrangement T 3# O P8 M 9? O P ?5: 99? Z ?5: equals ½ of 1660 written in decimal numbering system. [Notice the two designations: 6(10)6 for nucleon number and 1660 for atom number express a specific self-similarity.]
166 (6).. 732 G (6).. 832 G (6).. 932 .1353 .1553 .1753 (5) 621 (5) 721 (5) 821 1131 1331 1531 (4) 510 (4) 610 (4) 710 909 1109 1309 (3) 399 (3) 499 (3) 599 687 887 1087 (2) 288 (2) 388 (2) 488 465 665 865 (1) 177 (1) 277 (1) 377 (111) 243 (111) 443 (111) 643 (1) 066 (1) 166 (1) 266 021 221 421 (2) -045 (2) 055 (2) 155 -201 -001 199 (3) -156 (3) -056 (3) 044 -423 -223 -023 (4) -267 (4) -167 (4) -067 -645 -445 -245 (5) -378 (5) -278 (5) -178 -867 -667 -467 (6) -489 (6) -389 (6) -289 -1089 -889 -689 (7) -600 (7) -500 (7) -400
Survey B.7. The arithmetical system which is in relation with the system, presented in Survey (7 3+ O"#R P >54 ^ ??aC O$ R P column 1089 M 200, an ' OP >54 M 400.)
167 N a1, a2 A D d1 d2 (4) 999 499 1887 887 776 1111 111 (3) 888 388 1665 665 554 1111 111 (2) 777 277 1443 443 332 1111 111 (1) 666 166 1221 221 110 1111 111 (0) 555 055 999 (-1) 444 777 (-2) 333 555 (-3) 222 333 (-4) 111
Survey B.8. An insert from Survey B.6; N: the numbers in relation to nucleon number 1443; a1, a2: the numbers in relation to atom number 166 and 277, respectively; A: the numbers in relation 99?C "= "#R number 332 as the total number of O P I F " 3 68C 1: all differences in relation to the difference 1443 M 332 = 1111; d2: all differences in relation to the difference 443 M 332 = 111.
168 Appendix C
G S P T A D L E V N I Q C K M R F H Y W
(G, P) (A, L, V, I) (C, M, F, Y, W, H) (R, K, Q, N, E, D, T, S)
Figure C.1. Four diversity types of protein amino acids: 2 AAs with non-standard and 4 AAs with standard hydrocarbon side chain; then 6 AAs with different, and 8 with the same O PFP /P = ' # M through the principles of minimum change and continuity M follows a new arrangement, such as in Figure C.2 (Rako64' 2011a, Fig. 2; 2011b, Fig. 2 on p. 822).
169 G 01 S 05 Y 15 W 18 39 78 A 04 D 07 M 11 R 17 39 102 24 C 05 T 08 E 10 F 14 37 13
102 N 08 Q 11 V 10 I 13 42 89 P 08 H 11 L 13 K 15 47 26 42 59 77 16 17 18 (1 x (2 x 68) 68) Figure C.2. A specific AA classification and systematization which follow from four diversity types (Figure C.1) in correspondence with a unique arithmetical arrangement (Table C.2). The ordering through the validity of two Mendeleev principles: minimum change and continuity (1, 5, 15, 18 of atoms in the first row), (1, 4, 5, 8 of atoms in the first column) (Rako64' 2011a, Fig. 1; 2011b, Fig. 3 on p. 828).
... (-2) ...... -22 (-1) -21 -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 (0) -10 -09 -08 -07 -06 -05 -04 -03 -02 -01 00 (1) 01 02 03 04 05 06 07 08 09 10 11 (2) 12 13 14 15 16 17 18 19 20 21 22 (3) 23 24 25 26 27 28 29 30 31 32 33 (4) 34 35 36 37 38 39 40 41 42 43 44 (5) 45 46 47 48 49 50 51 52 53 54 55 (6) 56 57 58 59 60 5B 62 63 64 65 66 (7) 67 68 69 70 71 72 6D 74 75 76 77 (8) 78 79 80 81 82 83 84 7F 86 87 88 (9) 89 90 91 92 93 94 95 96 97 98 99 (A) A0 A1 A2 A3 A4 A5 A6 A7 A8 A9 AA (B) B1 B2 B3 B4 B5 B6 B7 B8 B9 BA BB Table C.1. The Table of minimal addition in decimal numbering system. A specific arrangement of natural numbers in decimal numbering system, going from 01 to 11 and so on (Rako64' 2011a, Tab. 4; 2011b, Tab. 4 on p. 826).
170 26 = 26 26 + 42 + 59 + 77 = Y 16 +17 + 18 = Z 26 + 16 = 42 Y = 204 Z = 51 42 + 17 = 59 Y/4 = 51 Z = Y/4 59 + 18 = 77
Survey C.1. The unique arithmetical relations which follow from the system presented in Table C.1 (Rako64' 2011a, Equations 4.1; 2011b, Equations 3 on p. 826).
2 x1 + y1 = 36 = 6 (x1 = 26; y1 = 10) 2 x2 + y2 = 25 = 5 (x2 = 17; y2 = 08) 2 x1 M y1 = 16 = 4 2 x2 M y2 = 09 = 3
Survey C.2. The unique algebraic relations which follow from the system presented in Table C.1 (Rako64' 2011a, Equations 4.2; 2011b, Equations 4 on p. 827).
171 ... (-2) ...... -22 (-1) -21 -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 (0) -10 -09 -08 -07 -06 -05 -04 -03 -02 -01 00 (1) 01 02 03 04 05 06 07 08 09 10 11 (2) 12 13 14 15 16 17 18 19 20 21 22 (3) 23 24 25 26 27 28 29 30 31 32 33 (4) 34 35 36 37 38 39 40 41 42 43 44 (5) 45 46 47 48 49 50 51 52 53 54 55 (6) 56 57 58 59 60 61 62 63 64 65 66 (7) 67 68 69 70 71 72 73 74 75 76 77 (8) 78 79 80 81 82 83 84 85 86 87 88 (9) 89 90 91 92 93 94 95 96 97 98 99 (10) 100 101 102 103 104 105 106 107 108 109 110 (11) 111 112 113 114 115 116 117 118 119 120 121
Table C.2. This Table is the same as Table C.1, except the first, highlighted column and the left diagonal, so that the following law is to be detected: the left diagonal appears as the sum of all >' # ^ >' ' ' G
172 01(-10) -9 -8 -7 -6 -5 -4 -3 -2 -1 0 21 32 43 54 65 76 87 98 109 120
12 012 024 036 048 060 072 084 096 108 120 (220) 241 252 263 274 285 296 307 318 329 340
23 253 276 299 322 345 368 391 414 437 460 (220) 461 472 483 494 505 516 527 538 549 560
34 714 748 782 816 850 884 918 952 986 1020 (220) 681 692 703 714 725 736 747 758 769 780
45 1395 1440 1485 1530 1575 1620 1665 1710 1755 1800 (220) 901 912 923 934 945 956 967 978 989 1000
56 2296 2352 2408 2464 2520 2576 2632 2688 2744 2800 (220) 1121 1132 1143 1154 1165 1176 1187 1198 1209 1220
67 3417 3484 3551 3618 3685 3752 3819 3886 3953 4020 (220) 1341 1352 1363 1374 1385 1396 1407 1418 1429 1440
78 4758 4836 4914 4992 5070 5148 5226 5304 5382 5460 (220) 1561 1572 1583 1594 1605 1616 1627 1638 1649 1660 89 6319 6408 6497 6586 6675 6764 6853 6942 7031 7120
Table C.3. The Table follows from Table C.2 with the multiplication of all neighbouring pairs in first column (the numbers on the diagonal), of their predecessors (the numbers for the diagonal) and of their successors (the numbers after the diagonal). The differences increase by 11, and the differences of differences by the twentieth multiple of 11, the number 220, which is the first friendly number. Here one must notice that the numbers on the left diagonal are the same numbers which appear in the arithmetical system presented in Figure 4.
173 1st 2nd letter 3rd lett. lett. U C A G 00. UUU 08. UCU 32. UAU 40. UGU U 01. UUC F 09. UCC 33. UAC Y 41. UGC C C U 02. UUA 10. UCA S 34. UAA 42. UGA A CT 03. UUG L 11. UCG 35. UAG CT 43. UGG G W 04. CUU 12. CCU 36. CAU 44. CGU U 05. CUC 13. CCC 37. CAC H 45. CGC C C 06. CUA L 14. CCA P 38. CAA 46. CGA R A 07. CUG 15. CCG 39. CAG Q 47. CGG G 16. AUU 24. ACU 48. AAU 56. AGU U N S 17. AUC I 25. ACC 49. AAC 57. AGC C A 18. AUA 26. ACA T 50. AAA 58. AGA A 19. AUG 27. ACG 51. AAG KR59. AGG G M 20. GUU 28. GCU 52. GAU 60. GGU U D 21. GUC 29. GCC 53. GAC 61. GGC C V A G G 22. GUA 30. GCA 54. GAA 62. GGA A E 23. GUG 31. GCG 55. GAG 63. GGG G Table C.4. The standard Genetic Code Table. This Table represents the relations within the so O $ P # four diversity types of protein amino acids and corresponding codons: the first (italics) and the second type without tones, but the third and the fourth in tones: light third type and dark the fourth one. The codon number: for the first type 08, the second 17, the third 10 and the fourth 26, just as in algebraic system in Survey C.2 (Rako64' 2011a, Fig. 3; 2011b, Tab. 6 on p. 829).
174 Appendix D
Figure D.1. The full generalization of Golden mean through Fibonacci sequences, in relation to natural numbers series (arXiv:math/0611095v2 [math.GM], Section 6).
175 G 4 4 5 4 (13) 3 3 4 3 (10) 2 2 3 2 (7) 1 1 2 1 (4) 0 1 1 1
Figure D.2. The "evolution" of a triadic Cantor set (the simplest possible fractal), placed in the zeroth position; the evolution through the divergence for one unit in all three positions. From the first position onwards there is an "Extended triadic Cantor set" through the number of quantities at levels. Here a paradoxical situation becomes obvious: the farther we move from the beginning, the closer to it we get!? The biological meaning could be this: after a million years since the origin of life on Earth there were a lot of different species of organisms, but one and the same genetic code; after a hundred million years even a greater number and a greater variety of the species existed and the code remains the same; After a billion years everything is still enormously increased, but the code remained the same. The third case (dark tones) corresponds # "#R ' 9? ' designated on the right of Figure (4-7-10-13- G8 # 0 )R / = OH I >?:P / # I ^ 9' :' >' ?' G
176 Figure D.3. The visualization of the Cantor triadic set as an infinite binary tree.
177 Appendix E
(a) 49 74 V10 L13 C05 E10 Q11 M11 I13 R17 W18 Y15 G01 A04 S05 D07 N08 T08 P08 K15 H11 F14 25 56
74 (56) 130 (c) (222 / 221) (b) 62 91 30 56
92 (55) 147
(d) 166 (111) 277 (443) (e) 113 066 47 111 (Tab.4.1) (Tab. 4.2) 66 177 (f) 24 (32) 66 (Tab.4.1) 40 (32) 47 (Tab.4.1)
64 113 (g) (Darwin code) 233 / 443 (Gen. code) (443 M 332 = 111)
Table E.1. (a) The first class of AAs is in the upper row, and in the lower row there is the second 3 )64' 44:8= O # / # ;/ (Class II with 81 and Class I with 123 atoms.) The ten amino acid pairs, natural pairs from the chemical aspect, are classified into two classes. Class I contains larger amino acids (larger within the pairs), all handled by class I of enzymes aminoacyl-tRNA synthetases. Class II contains smaller amino acids, all handled by class II / G # atoms within side chains of class II AAs (given here as index); from left to right: first there are
178 ' GA+ D-Y is simpler as only aromatic and H- W is more U /BP 3 )64' > ' 8 0 )R account of nucleon number within the amino acid constituents of GC, in their side chains (Figure 1.1) is as follows: [1 x (G1+A15+ P41+ V43+ T45 + C47 + I57+ N58 + D59 + K72 + Q72 + E73 + M75 + H81 + F91 + Y107 + W130)] + [2 x (S31 + L57 +R100)] = 1443. J 0 )R account is done, with an iteration more, for the number of atom, the result is as follows: [2 x (G1 + A4 + C5 + D7 + N8 + T8 + P8 + E10 + V10 + Q11 + M11 + H11 + I13 + F14 + Y15 +K15 + W18)] + [3 x (S5 + L13 + R17)] = 0443 (here: row d). G ' # "#R diagram there are the next "branch" entities/quantities: 276 branches plus 46 nodes + 10 branchings, in total 332. The significant differences are as follows: 1443-332 = 1111and 443-332 = 111, both determined by the unity change law (here: row g); (b) Atom number within 23 amino acid molecules as in (a), except that two-meaning AAs (L,S,R) participate twice in the account: 204 + 35 = 239 = 92 + 147 38 ` O = :9 Z 9: ^ ?> Z 92 = 222; (d) The result of two sumation: 74 + 92 = 166 and 130 + 147 = 277; (d) The sumation # = 77 Z :: ^ 99?C 38 "#R 9 d 9C 38 "#R 377 M 64 = 2) as in Tables 4.1 in correspondence with two results in genetic code: 92 M 91 = 1 and 74 M 74 = 0; (g) Final result in GC (443) in relation "#R 3??8' ) 96 and 5.
179 Appendix F. A simple syllogism
"#R # the presented arithmetical / algebraic systems 2. Genetic code corresponds with presented arithmetical / algebraic systems ? ' "#R # the Genetic code
Distrib. of AAs after Cloister energy and atom number Relations Chemical pairs H 0.00 1.46 K xx H 0.00 1.46 K GGGGG (H M W) 44 A -0.09 0.91 Q 60 57 A -0.09 0.91 Q 68 (A M G) 45 G -0.16 0.87 R 45 54 G -0.16 0.87 R 54 V M L 89 W -0.25 0.71 E 105 111 W -0.25 0.71 E 122 (K M R) V -0.52 0.69 D V -0.52 0.69 D (44+44 = 88)49 Q M E 194 L -0.54 0.52 N (60+56 = 116)50 D M N L -0.54 0.52 N (233)51 56 I -0.56 0.46 P 44 I -0.56 0.46 P (I M P) 45 F -0.56 0.42 Y 45 43 F -0.56 0.42 Y 36 F M Y 101 M -0.57 0.27 T 89 36 M -0.57 0.27 T 36 (M M T) 79 72 C M S C -0.73 ..0.24 S C -0.73 ..0.24 S 125 (102+23) 125 = 57+68 190 151 79 (102-23) 79 = 43 +36 Odd 46 (102-1) 55 102±x (For x = 23 we have the Even.. 54 (102+1) 49 correspondence with 276)
Table F.1. Distribution of amino acids after Cloister energy (Swanson, 1984) and atom number.
[Note F.1. The chemical pairs after (Dlyasin, 1998, 2011; Rako64' 445' Survey 4, p. 290; Rako64' 2004, Figures 1 and 2, p. 222). The pairs G-A and V-L as well as S-T and C-M after Dlyasin; in a vice versa logic: G-V and A-L as well as S-C and T-M after Rako64; all other is the same].
49 # :7 = 3:78 ^ 55 3>55Z 55 ^ :78 50 # 7> 67 = 3 78 ^ 67 367Z7> ^ 78C 56 as all primary branches at 1-10 levels as well as at 11-14 levels, for two species A and I (Tab. 2.1); the 60 as total number of all branches in upper part of Darwin Diagram (DD): the 56 as said, plus 4 branches in second set of species (9 species) as it is shown in (Table 6.1). [Note: in Table 6.1 see above the levels 10-14 with only 4 branches.]; the 116 as complement of 216 (footnotes 55 and 58). 51 The 233, as all branches (prim. + second.) for two species, "A" and "I" into (3 x 5) levels (Tab. 4.5). Here: 111 + 122 equals 233. In DD: 112 as all primary branches + 121 as all secondary branches equals 233.
180 (111+ 01 = 112)52; (122 M 01 = 121)53; [89 + 89 = 178; (178 M 01 = 177)54 (101 + 105 = 206 (206 + 10 = 216)55; (206 + 178 = 384)56 (216 + 177 = 393); (384 + 393 = 777)57 (116 + 216 = 332);58 (88 +188 = 276)59 (190 M 151 = 39)60
(233 M 194 = 39)
Survey F.1. Relations between quantitatives of Genetic code and existing quantitatives within "#R 3J8
M 61 (194 151 = 43) 44 + 44 = 88 [(233) + (43)62 = (276)] (233 M 190 = 43) 60 + 56 = 116
Survey F.2. Relations between quantitatives of Genetic code and existing quantitatives within "#R 3J8
52 The 112 as the number of all primary branches for two species, "A" and "I" into (3 x 5) levels (Tab. 2.1). 53 The 121 as all secondary branches for two species, "A" and "I", into (1-10) levels (Table 2.2). 54 The 177 as all branches (primary + secondary) for two species, "A" and "I" into (1-10) levels (Tab. 4.1). xxx 55 The 216 as all branches (primary + secondary) for all 11 species at 1M10 levels (Table 7.2). xxx 56 The 384 > ' # O P O P 57 Cf. with the starting 777 in Table B.2. 58 The 216 as in footnote 55; then the 116 contains all other quantitatives to the sum of 332 (Table 5) O P quantitatives: 60 branches at 11-14 levels into all 11 species, plus 46 nodes (Table 3.1), plus 10 branchings (Table 3.2) [Note: the nodes and branchings exist only in species, "A" and "I".] 59 The 88 as all primary branches for all 11 species at 1M10 levels (Table 7.5). The 188 as the sum of all other branches to the total sum of 276. 60 The 39 as all branches (primary + secondary) for 9 species (B, C, D, E, F on the left and G, H, K, L on the right) at 0-9, i.e. 1-10 levels (Table 6.5). 61 The 43 as all branches (primary + secondary) for 9 species (B, C, D, E, F on the left and G, H, K, L on the right) at 0-14 levels (Table 6.4). 62 The 43 as in footnote 61. 181 Appendix G. The number of hydrogen bonds within the set of four nucleotides
Why 2-3 and not 1-2 hydrogen bonds within the set of four nucleotides (UA connected with two and CG with three hydrogen bonds)? The answer follows from the relationships presented in Tables G.1 and G.2. If we have the alphabet of four letter (UCAG in Table G.1), then there are six their pairs (UC, AG, UA, CG, UG, CA). Also there are two possibilities for bonding (Tables G.1 & G.2). Going from the arrangement in Table G.1 to the arrangement in Table G.2 the pairs UG, CA appear to be invariant, but other four (two and two: UC/AG and UA/CG) variant. By this, from the chemical aspect we must speak: 2 original pairs (UC/AG or UA/CG), 6 derived pairs (UC, AG, UA, CG, UG, CA), 10 hydrogen bonds (5+5 or 4+6). Altogether this is the # [ )R ' N = 2(2n+1) (n = 0,1,2,3) as it is presented in Section 4.3 and Box 2. In Table G.1 we can find this arrangement only it the case with 2-3 hydrogen bonds, in wich case the principles of continuity and minimum change are also valid. One must notice that the pattern 4-5-5-6 of hydrogen bonds corresponds with the same / U # R eotide doublets. (Cf. Tables 1 & 2 in Book of Abstracts M Theoretical Approaches to Bioinformation Systems,TABIS 2013, 17-22 September 2013, Belgrade, Serbia.) (Proceedings in press.) Notice also that this pattern corresponds with the 4-5-5-6 amino acid pairs, presented in this chapter in Table F.1.
182 CG 8 CG 6 CG 4 CA 7 14 CA 5 10 CA 3 6 UG 7 14 UG 5 10 UG 3 6 UA 6 UA 4 UA 2
2 2 2 2 2 2 2 2 2 2 C 2 2 G 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2 2 U A
2 2
N = 2(2n +1) 0 0
+ ^ ' 7' >' 9G CG 2 CA 1 2 UG 1 2 UA 0
Table G.1. The number of real and hypothetical hydrogen bonds (I)
183 AG 7 AG 5 AG 3 CA 7 14 CA 5 10 CA 3 6 UG 7 14 UG 5 10 UG 3 6 UC 7 UC 5 UC 3
2 2 2 2 2 2 2 2 2 2 C 2 2 G 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2 2 U A
2 2
N = 2(2n +1) 0 0
+ ^ ' 7' >' 9G AG 1 CA 1 2 UG 1 2 UC 1
Table G.2. The number of real and hypothetical hydrogen bonds (II)
184 Appendix H. $# #' +R*
[Darwin, Ch. (1876) The Origin of Species, John Murray, 6th ed., with additions and corrections, London]
Citation 1a (p. 90). The accompanying diagram G
O G G P (Explanation 1a)
Explanation 1a. It is clear that Darwin here insists on the letters as such. In this case in the question are large letters below, under the zeroth level.
Citation 1b (p. 92). In our diagram G
OG by G P 3TU )
Explanation 1b. At the levels 1-10 all varieties are designated O/ P However, at the levels 11-14, only varieties at the level 14 are with letters and other (3 x 14 = 42 varieties) at 11-13 levels are without letters (see Citations 2 & 3).
185 Citation 2 (p. 91): The intervals G
O thousand Gwell-marked P (Explanation 2)
Explanation 2. Varieties as the primary branches, and as the letters to the 10th level, at the same time; well-marked also at the level 14. H.1
Citation 3 (p. 91): In the diagram ... a condensed and simplified form
O G fourteen-thousand P (Explanation 3)
Explanation 3. J O fourteenP' th , 12th and the 13th at the same time.
** From this small discussion it follows: from all primary branches (varieties) in two species 3`O `JO8' ' ; # ' ' :> 36 U 98 `well- ) P / :> 9 3? U 98 ) # 3 e 56 primary branches on the levels I-X, marked with the letters, plus 14 on the 14th level, equals 70 `well- ) P / C :> Z 9 ^ 8 On the other hand, from all primary branches (varieties) in remaining 9 species, them 36, organized without of two trees, and presented in Table 6.1, the 3 (1 x 3) are well-marked by 3 letters (E & F at the 10th level and F at 14th level) and 33 (11 x 3) are not marked with the letters. Altogether, 70 + 3 = 73 well-marked and 42 + 33 = 075 not marked (75 + 73 = 148 as in Table 7.3). (Cf. first two cases in last columns of Tables H.1, H.2 and H.3). By this one must notice that the result 075 corresponds with the middle position in Table H.1 (075 vs 185). Also, from these relationships it follows t O:?P # :? :? / # 0 )R 3.' 0' # 8 )
186 the sum of all branch-letters entities, that means: 73 + 148 = 221, which result we have in the GC 3 T ' ''= 9: Z :9 ^ 4 Z ?> ^ 8' # 0 )R 3# iteration more) also acts. On the other hand, the 73 well-marked primary branches and 75 non- marked, appear to be in the ratio 74±1 (Table H.3), what a balance is and where the 74 is also a 0 )R # O P ** In addition to the said: there is a whole Darwin Diagram Space, i.e. the Darwin Diagram 0/ 3""08' O- P 3 `O < `JO8 O-tree- P [left part (B,C,D,E,F) & right part (G,H,K,L)]. Within whole DDS there are 73 letters, and within O- P / 3 8 D ' ) calculate the sum 73 + 112 = 185. Why? Because the 73 are branch-letters in whole DD system, and 112 are branches in DD sub-system; the sub- / ` O # / A`M O ` O ; ` / / O (Marcus, 1989, p. 103).] #' # [ ' 56 ` O ' middle position within a specific arithmetical system, the system of relationships between 5 0 )R `H I ?:O
M # ""0 95 / 3 :?8' # O- P 3 ` O `# O8 :> 3:? T < D > D 14th level); the sum of 148 primary branches in both spaces (tree-space and non-tree-space as a whole) plus 70 letters in tree- 3 ` O8 ^ 5C 5 I 3 # # 8' minus 33 primary non- ) 3 # ` O8 I 56 3+= ?? primary non- ) # # O-tree- P' # ? ?7 Table 6.1; these 3 are M the 3 well-marked primary branches, designated with E & F at 10th and F at 14th level.)
187 Multiples of 14 & 37 Sums Double values 1 14 (23) 37 51 28 (46) 74
2 28 (46) 74 102 56 (92) 148
3 42 (69) 111 153 84 (138) 222
4 56 (92) 148 204 112 (184) 296
5 70 (115) 185 255 140 (230) 370
6 84 (138) 222 306 168 (276) 444
7 98 (161) 259 357 196 (322) 518
8 112 (184) 296 408 224 (368) 592
9 126 (207) 333 459 252 (414) 666
Table H.1 # 5 0 )R OH I ?:P re some GC significant numbers: 204 as the number of atom within 20 amino acid side chains; the number 0255 corresponds with the total number (1255) of # > C :9 # O P (Notice an analogy between this pattern 0255 / 1255 and the pattern 0443 / 1443 in Table E.1, valid for the number of atoms and nucleons, respectively.)
188 Tab. H.2 1 2 3 4 5 6 3+6 I (24 18. 42)63 G (24. 09. .33)64 .. 7565 II (44 26 70)66 (03 00 03)67 7368 I+II (68 44 112)69 (27 09 36)70 14871 III (77 44 121)72 (03 04 07)73 12874 I+II+III (145. .88 .233)75 (30 13 43)76 27677
Table H.2. Distributions and Distinctions within Darwin Diagram, DDDD-1= O- P 3 ''?8 O-tree- P 39'6'78 >-14 levels; I.1: The left tree- 3OP): the 24 non- marked primary branches (3 x 8 = 24) on 11-13th levels; I.2: The right tree- 3OJP8: the 18 non-marked primary branches (3 x 6 = 18) on 11-13th levels; I.3: The sum of previous two
63 The results 24-18-42 (non-marked primary branches on 11-13th levels) (non-marked by the small letters) in relation to total primary branches on both trees (A & I): 68-44-112 (Table 2.1). 64 The results 24-09-33 (non-marked primary branches on 0-14th levels) (non-marked by the small letters) in relation to total primary branches at the left and right non-tree-space, respectively (left: B,C,D,E,F; right: G,H,K,L): 27-09-36 (Table 6.1). 65 The total number of non-marked primary branches (varieties non-designated with small letters). 66 The results 44-26-70 as well-marked primary branches within tree- ' OP OJP 3 Figure 5). 67 The results 03-00-03 as well-marked primary branches within non-tree-space (only within the left space: B,C,D,E,F; that, because within the right space: G,H,K,L the well-marked primary branches do not exist (Table 6.1). 68 The total number of well-marked primary branches (varieties designated with small letters). 69 The total number of primary branches within tree- ' OP OJP 3 D 68 70 The total number of primary branches within non-tree-space (left: B,C,D,E,F; right: G,H,K,L) (Table 6.1). At the = "#R I 7 D 7 71 The total number of primary branches within tree- 3OP < OJP8 -tree-space (left: B,C,D,E,F; right: G,H,K,L). The number 148 together with the subsequent number below (128) appears to be a change for ±10 in relation to the half of number 276 as total number of branches within the Diagram (2 x 138 = 276) (cf. Table 7.3 with Table 7.1). 72 The total number of secondary branches (non-varieties) (Table 2.2 and Figure 4). 73 All secondary branches for 9 species (B, C, D, E, F on the left and G, H, K, L on the right) at 0-9 levels (Table 6.3). 74 The total number of secondary branches within tree- 3OP < OJP8 -tree-space (left: B,C,D,E,F; right: G,H,K,L) (Table 6.1). The number 128 together with the subsequent number above (148) appears to be a change for ±10 in relation to the half of number 276 as total number of branches within the Diagram (2 x 138 = 276) (cf. Table 7.4 with Table 7.1). The number 128 as the sum of 121 + 7. 75 The total number of branches (primary plus secondary) within tree- 3OP < OJP8 3 96 < 6C in relation to Table E.1, g). 76 The total number of branches (primary plus secondary) within non-tree-space (left: B,C,D,E,F; right: G,H,K,L) at 0-14 levels (Table 6.4). 77 The total number of branches (primary plus secondary) within tree- 3OP < OJP8 -tree-space (left: B,C,D,E,F; right: G,H,K,L) (Table 7.1). The 276 as the sum of 233 + 43. 189 (Table 2.1); I.4: The left non-tree-space (B,C,D,E,F): the 24 non-marked primary branches (21 on the levels 0-9 and 3 on the levels 11-13); I.5: The right non-tree-space (G,H,K,L): the 9 non- marked primary branches (all 9 on the levels 0-5); I.6: The sum of previous two (Table 6.1); II.1: The left tree- 3OP8= 99 #-marked primary branches (36 on the levels 0-9 and 8 on the level 14); II.2: The right tree- 3OIP8= 26 well-marked primary branches (20 on the levels 0-9 and 6 on the level 14); II.3: The sum of previous two (Table 2.1); II.4: II.4: The left non-tree-space (B,C,D,E,F): the 3 well-marked primary branches (E & F with the start on the level 9th and the finalization on the level 10th; plus F with the start on the level 13th and the finalization on the level 14th); II.5: The right non-tree-space (G,H,K,L): without well-marked primary branches; II.6: II.6: The sum of previous two (Table 6.1). With all these facts it must be noted that the vertically shaded field is in connection with arithmetical systems in Figures 4 & 5, and the horizontally shaded field # "#R I' observed in Table 6.1 and presented in Figure 6.
Tab.H.3 1 2 3 4 5 6 3+6 I 24 24 48 18 09 27 75 ±74 II 44 03 47 26 00 26 73 I+II 68 27 95 44 09 53 148 III 77 03 80 44 04 48 128 I+II+III 145 30 175 88 13 101 276
Table H.3. Distributions and Distinctions within Darwin Diagram, DDDD-2: Left part of Darwin Diagram (DD) versus Right part, 1-14 levels. All data in columns 1, 2, 4, 5 are the same as in Table H.2, except the interchange of two columns: 2 & 4, respectively. The horizontally # "#R I [9 [6C The vertically shaded field is in connection with a unique situation in a specific arithmetical system, presented in Table H.6.
Table H.4 -09 -09 -18 -18 1 09 09 18 2 00 09 09 -09 09 00 G 2 18 09 27 3 09 09 18 00 09 09 3 27 09 36 4 18 09 27 09 09 18 4 36 09 45 5 27 09 36 18 09 27 5 45 09 54 6 36 09 45 27 09 36 6 54 09 63 7 45 09 54 36 09 45 7 63 09 72 8 54 09 63 45 09 16 G 8 72 09 81 9 63 09 72 54 09 36 9 81 09 90 10 72 09 81 63 09 52 G
190 Table H.4. / 4 # "#R equation in the darker ton (Variant I).
Table H.5 -1 -1 -2 -2 1 09 09 18 2 08 09 17 07 09 16 G 3 27 09 36 4 26 09 35 25 09 34 5 45 09 54 6 44 09 53 43 09 52 7 63 09 72 8 62 09 71 61 09 70 9 81 09 90 10 80 09 89 79 09 88 G
Table H.5. An arithmetical system of multiples of number 9 with an inclusion of DarwinR equation in the darker ton (Variant II).
2,2,2 10 25 35 2,2,2 10 25 35 2,2,2 10 25 35 2,2,2 20 35 55 2,2,2 20 35 55 2,2,2 20 35 55 2,2,2 30 45 75 2,2,2 30 45 75 2,2,2 30 45 75 2,2,2 40 55 95 2,2,2 40 55 95 2,2,2 40 55 95 2,2,3 50 65 115 2,2,3 50 65 115 2,2,3 50 65 115 2,2,3 60 75 135 2,2,3 60 75 135 2,2,3 60 75 135 2,2,3 70 85 155 2,2,3 70 85 155 2,2,3 70 85 155 2,2,3 80 95 175 2,2,3 80 95 175 2,2,3 80 95 175 2,3,3 90 105 195 2,3,3 90 105 195 2,3,3 90 105 195 3,3,3 100 115 215 3,3,3 100 115 215 3,3,3 100 115 215 3,3,3 110 125 235 3,3,3 110 125 235 3,3,3 110 125 235 3,3,3 120 135 255 3,3,3 120 135 255 3,3,3 120 135 255 3,3,3 130 145 275 3,3,3 130 145 275 3,3,3 130 145 275 3,3,3 140 155 295 3,3,3 140 155 295 3,3,3 140 155 295 3,3,3 150 165 315 3,3,3 150 165 315 3,3,3 160 175 335 3,3,3 160 175 335 3,3,3 170 185 355 3,3,3 180 195 375
Table H.6. A specific arithmetical system with two or three digit numbers. In relation to left and right area, the middle area appears to be unique: the central row (80-95-175) is in a direct connection with only one situation where appears a digit-triplet with the Gray code changes (2-2- 3 / 2-3-3 / 3-3-?8 * "#R [? 3+ the correspondence with Table H.1 from the aspect of appearance of two-three digit numbers.)
191 Appendix K. Another correspondence with the genetic code78
From Table K.1 follow Tables K.2 and K.3 with first half of AAs in first column and the second one in the second column. For each AA is given atom number in whole molecule. As we can see, atom number within 12 AAs in second column is 233; exactly as the number of all 3 / /8 # ' < J' # "#R 3 968C in first column 233 M 1 = 232.
1st 2nd letter 3rd lett. U C A G lett. 00. UUU F 08. UCU 32. UAU Y 40. UGU C U 01. UUC 09. UCC 33. UAC 41. UGC C U 02. UUA L 10. UCA S 34. UAA CT 42. UGA CT A 03. UUG 11. UCG 35.UAG 43. UGG W G 04. CUU 12. CCU 36. CAU 44. CGU U H 05. CUC 13. CCC 37. CAC 45. CGC C C L P R 06. CUA 14. CCA 38. CAA 46. CGA A Q 07. CUG 15. CCG 39. CAG 47. CGG G 16. AUU 24. ACU 48. AAU 56. AGU U N S 17. AUC I 25. ACC 49. AAC 57. AGC C A T 18. AUA 26. ACA 50. AAA 58. AGA A K R 19. AUG M 27. ACG 51. AAG 59. AGG G 20. GUU 28. GCU 52. GAU 60. GGU U V D 21. GUC 29. GCC 53. GAC 61. GGC C G A G 22. GUA 30. GCA 54. GAA 62. GGA A V E 23. GUG 31. GCG 55. GAG 63. GGG G
Table K.1. `The Table of the standard genetic code (GCT). Total codon space is divided into three parts in correspondence with the harmonic mean (H) of the whole codon space sequence (a) and its half (b), where a = 63, b = 31.5 and H = 42. ... In the central area, the three stop codons (CT, codon terminations) O ( )64' > ?' 8 (/ # self- similarity between patterns of quantitives: 022 /021 of codons in this Table versus 122 / 121 of atoms in Table K.3.
78 Appendices labeled with the letters "I" and "J" in this chapter do not exist. 192 F 23 20 H F H L 22 20 Q L 45 40 Q L 22 17 N L N 41 I 22 24 K I 44 K M 20 16 D M D V 19 19 E V 39 35 E V 19 14 C V C 41 S 14 27 W S 33 W P 17 26 R P R 34 40 T 17 14 S T S A 13 26 R A R 36 Y 24 10 G Y 37 G 114 119 233 118 115 233 118 114 232 114 118 232 232 233 232 233
Table K.2. The Table follows from Table 1: odd /even atom number distinctions.
193 F 23 20 H L 22 20 Q (d)65 53(e) L 22 17 N 63 63 I 22 24 K M 20 16 D 244 M 233 = 011 V 19 19 E Crossing 128 + 116 = 244 (2 x 122) Horizontal 118 + 126 = 244 (2 x 122) Vertical 118 + 126 = 244 (2 x 122) V 19 14 C S 14 27 W (f)49 66(g) P 17 26 R 55 51 T 17 14 S A 13 26 R 232 M 121 = 111 Y 24 10 G 100 + 122 = 222 232 233 C 100 + 121 = 221 (1 x 221) H 115 + 106 = 221 (1 x 233 M 232 = 001 221) V 104 + 117 = 221 (1 x 221) 232 M 122 = 110 384 M 100 = 284
Table K.3. The Table follows from Table K.2. For details cf. Surveys K.1 and K.2.
194 n-VPA (49) + p-WSG (51) = 100 (The reference sequence)
FLM (a) HND FLM (d) QKE
LIV QKE LIV (e) HND
VPA (b) STY VPA (f) WSG
CWG (c) RSR STY (g) CRR
Survey K.1. This survey is the key for the reading and understanding of amino acid arrangement in Tables K.2 and K.3, in odd/even positions, respectively. Also, after our hypothesis (and prediction) M the key of positioning and hierarchy of AAs within proteins. By this the key of the key is the reference sequence (above) with the unity of arithmetical and physico-chemical balance: 50±1 of atoms in non-polar and polar sub-sequence, respectively; balance-nuancing in polarity/nonpolarity: Valine & Alanine as nonpolar; Proline as nonpolar in polar requirement 3M ' ' 477C LR ) ' 4:68 / / U 3L/ and Doolittle, 1982) and cloister energy (Swanson, 1984); Serine as polar; Tryptophan and Glycine as polar in hydropathy index, and nonpolar in cloister energy. Except this way, the balance-nuancing is evident through realization of three logics: (a) nonpolar AAs on the left and polar on the right; (b) AAs in odd positions are nonpolar and AAs in even positions are polar; (c) outer AAs, CWG, are nonpolar (in cloister energy) and inner, RS, polar. The arrangement on the right, the sequences d-g, in correspondence with their arithmetical solutions in Table K.3, shows further nuancing and unity of arithmetical balances and polarity / nonpolarity.
/00 - 07/08 - 15/16 - 23/24 - 31//32 - 39/40 - 47/48 - 55/56 - 63/ Survey K.2. OThe determination of 28 92 156 220 284 348 412 476 the series of the numbers 0-63. When 64 64 64 64 64 64 64 we look closely into the structure of /00 - 07/00 - 15/00 - 23/00 - 31//00 - 39/00 - 47/00 - 55/00 - 63/ the sequence 0-63 of the series of the 28 120 276 496 780 1128 1540 2016 natural numbers we come to the 92 156 220 284 348 412 476 obvious and self-evident explanation of the reason why the genetic code must be six-bit code, no matter if it is the manifestation in the form of the Gray Code model (Swanson, 1984, p 188), or it is in the form of the Binary 3 )64, 1994, p. 38). There must be 8 codon, i.e. amino acid classes. The structure of the sequence 0-63 is strictly determined by third perfect number (496) and the sum consisted of the first pair of the friendly numbers (220+2848P 3 )64' 497b, p. 60). [Cf. with the same pair of friendly numbers, in form 110-284, in Table K.3, below.]
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196 LR )' ( $C ' K ( 3 4:68 L )/ ) ) ) Dokl. Akad. Nauk. SSSR 223, 471M474. Marcus, S. (1989). Symmetry in the Simplest Case: the Real Line. Computers Math. Applic. 17, 103-115. -4' + 3> 8 + F F E )R quantum 037 as a base of (biological) coding/computing. Neuroquantology, 9, 702-715. Negadi, T. (2009) The genetic code degeneracy and the amino acids chemical composition are connected, Neuroquantology, Vol. 7, 1, 181-187; arXiv:0903.4131v1 [q-bio.OT]. Negadi, T. (2014) The genetic code invariance: when Euler and Fibonacci meet, Symmetry: Culture and Science, Vol.25, No.3, 145-288, 2014; arXiv:1406.6092v1 [q-bio.OT]. Pullen, S., 2005. Intelligent design or evolution? Free Press, Raleigh, N. Carolina, USA. )64' - - 3 44 8 ' Proceedings of Faculty of science (before: Faculty of philosophy), Chemistry Section, 2, 1-4' + )64, M. M. (1994). Logic of the Genetic Code, +6 )&' Belgrade. )64' --' 3 44:8 # /-tRNA synthetases in correspondence with the codon path cube. Bull. Math. Biol. 59, 645-648. )64' - -' (1997b) $ I / 0LE' + 3###) 8 )64' -- 31998a) The genetic code as a golden mean determined system, Biosystems 46, 283-291. )64' - - 3 445) Whole-number relations between protein amino acids and their biosynthetic precursors, J. Theor. Biol. 191, 463 M 465. )64' - - 32004) A harmonic structure of the genetic code, J. Theor. Biol. 229, 463-465. )64' - - 3>>58 $ E = D-Codon and Non-Four-Codon Degeneracy, arXiv:0802.1056v1 [q-bio.BM] )64' - - 3>>48 $ E = / determinism and pure chance, arXiv:0904.1161v1 [q-bio.BM]. )64' - - (2011a) Genetic Code: Four Diversity Types of Protein Amino Acids, arXiv:1107.1998v2 [q-bio.OT] )64' -- (2011b) Genetic code as a coherent system, Neuroquantology 9, 821-841. (www.rakocevcode.rs) )64' - - (2013) Harmonic mean as a determinant of the genetic code, arXiv:1305.5103v4 [q-bio.OT]. Shcherbak, V. I. (1993) Twenty canonical amino acids of the genetic code: the arithmetical regularities, part I, J Theor. Biol., 162, 399-401. Shcherbak, V.I. (1994) Sixty-four triplets and 20 canonical amino acids of the genetic code: the arithmetical regularities. Part II. J. Theor. Biol. 166, 475-477. Shcherbak, V. I. (2008) The arithmetical origin of the genetic code, in: The Codes of Life, Springer. Swanson, R. (1984) A unifying concept for the amino acid code. Bull. Math. Biol. 46, 187-207. Timiryazev, K. (1942) Istoricheskij metod v biologii, Akademiya nauk SSSR, Moskva.
197 Verkhovod, A. B. (1994) Alphanumerical divisions of the universal genetic code: new divisions reveal new balances, J. Theor. Biol., 170, 327M330. Woese, C.R. et al. (1966) On the fundamental nature and evolution of the Genetic Code. In: Cold Spring Harbor Symp. Quant. Biol. 31, 723M736.
198 SUPPLEMENTS
Supplement 1
Genetic Code as a Coherent System1
; R ' / of its coherence. Thus, it is shown that within genetic code there are four time four types of entities: four type of coding, four types of degeneracy, four stereochemical types of canonical amino acids and four types of diversity of the same these amino acids. In addition, it is revealed some harmonic structures arising through the determination by the golden mean, enzymes aminoacyl-tRNA synthetases, Gaussian algorithm and so on.
1. Preliminaries
This paper represents a summarizing review of our investigations of the genetic code (GC), but only in terms of coherence, that means in terms of the proofs that within the genetic code all its constituents appear to be in harmonic interconnections. As the first, there is the connection between amino acids (AAs) and codons, then of both and related arithmetical and/or algebraic regularities and so on. In order to show that the genetic code is a coherent system, we present, as the first, the fact that GC contains four times four entities: the four types of coding, four types of degeneracy, four stereochemical types of canonical AAs (Fig. 1) and four types of diversity of the same these AAs (Fig. 2). To understand how come the four types of coding one should start with the fact that GC works based on the connection between two alphabets, four-letter and twenty-letter, respectively. Those two cases appear to be optimal in terms of validity of the two principles, the principle of minimum change and continuity principle.
Moreover, regarding to the validity of these two principles the four-letter alphabet is not only optimal but also the one and only possible. Namely, from the four-letter alphabet follow three- letter words, each letter with two-letter root, and a single letter at the beginning. Altogether, we have the sequence of continuity: 4 - 3 - 2 M 1. If we would have a five-letter alphabet, from it generated words must be as four-letter, with the ambiguity in terms of the number of letters in the root - two or three letters? In six-letter alphabet the situation would be still more confused etc.
1 NeuroQuantology | December 2011 | Vol 9 | Issue 4 | Page 821-841.
201 Glycine (G) Proline (P) Valine (V)
Leucine (L) Alanine (A) Isoleucine (I)
Threonine (T) Serine (S) Cysteine (C)
Figure 1. The structure formulae of AAs in relation to four stereochemical types: Glycine, one and only within glycine stereochemical type; proline in proline type; valine and isoleucine within valine type, and all other within alanine stereochemical type.
As for the optimality of the number 20, there could be more reasons, but most important is this one which concerns to the generalization of the golden mean; such a generalization from which follows the so-called "metallic means family" (MMF) (Spinadel, 1999) as solutions of the Equation (1), for the various values of the parameters p and q:
(1) x 2 px q 0
202 In fact, if p = q = 1, we have the square equation of golden mean. For p = 2 and q = 1 we obtain the silver mean; for p = 3 and q = 1, the bronze mean etc. On the other hand, if p = 1 and q = 2, we obtain the copper mean; for p = 1 and q = 3, we get the nickel mean and so on. Special cases O P' I ^ ' 7' ' >' ?>' ' #hich solutions (x1, x2), given by TI 3 8' = 3 ' 8' 3' ?8' 3?' 98' 39' 68' 36' 78 D O / / / I #/P (Spinadel, 1999). What, however, is especially interesting, in connection with the said, is that Spinadel's integers (2, 6, 12, 20, 30, ...) appear in the diagonal of the very known Table of multiplication, with the number 20 in the central position (Table 1.1 in relation to Table 1.2). On the other hand, the number 20, together with all other essence GC-quantities (4, 16, 64, 81) can be found in HR / # 3U 2*' 4:>8 3 U8C relation to the total number of atoms in 20 AAs (384) (Tab. A.2).
G S P T A D L E V N I Q C K M R F H Y W
(G, P) (A, L, V, I) (C, M, F, Y, W, H) (R, K, Q, N, E, D, T, S)
Figure 2. Four diversity types of protein amino acids in a linear arrangement in form of the sequence 2-4- 6-8; then in a circular arrangement, in form of the sequence 5-5-5-5. From this last sequence it is possible a new arrangement in form of the sequence 4-4-4-4-4 as in system presented in Figure 3.
The following consequence of the validity of the two principles is that the coding must to be realized at the level of the quartet, quintet, sextet, etc., for the 4-letter, 5-letter, 6-letter alphabet, respectively. Hence, in the case of 4-letter alphabet inevitably there are four types of coding 3 8= J 3 8' JJ 3 8' JJJ 3? 8' J% 39 8' word from the quartet codes for one amino acid; then - two, three, four words for one amino acid, respectively. [Of course, it is possible that, except one whole quartet, for one amino acid code the words from another quartet too.]
203 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 0 2 4 6 8 10 12 14 16 18 0 3 6 9 12 15 18 21 24 27 0 4 8 12 16 20 24 28 32 36 0 5 10 15 20 25 30 35 40 45 0 6 12 18 24 30 36 42 48 54 0 7 14 21 28 35 42 49 56 63 0 8 16 24 32 40 48 56 64 72 0 9 18 27 36 45 54 63 72 81 Table 1.1. The multiplication Table of decimal numbering system. The harmonic multiplication Table of decimal numbering system O P' I ^ >' ' 7' ' >' ?>' 9' 67 72 on the diagonal in the form of doublets (pairs): 0-0, 2-2, 6-6, 12-12, 20-20, 30-30, 42-42, 56-56 and 72-72.
x2 + x = z z (0 x 0) + 0 = 00 0 x 1 = 00 (1 x 1) + 1 = 02 1 x 2 = 02 (2 x 2) + 2 = 06 2 x 3 = 06 (3 x 3) + 3 = 12 3 x 4 = 12 (4 x 4) + 4 = 20 4 x 5 = 20 (5 x 5) + 5 = 30 5 x 6 = 30 (6 x 6) + 6 = 42 6 x 7 = 42 (7 x 7) + 7 = 56 7 x 8 = 56 (8 x 8) + 8 = 72 8 x 9 = 72 (9 x 9) + 9 = 90 Table 1.2. The key of the harmonic multiplication Table. This key is related to positive integers: (0, 1),
(1, 2), (2, 3), (3, 4), (4, 5), (5, 6) which appear as solutions (x1, x2), given by Equation (1); as solutions for above given q (q = 0, 2, 6, 12, 20, 30, ...).
To understand that in the GC there are four types of degeneracy it is necessary the connection amino base - amino acid to be viewed from the aspect of the principle of donor M acceptor; from eight possible, four are implemented in the GC: I. (4), II. (3, 1), III. (2, 2), IV. (2, 1, 1); and four are not implemented: I. (1, 3), II. (1, 1, 2), III. (1, 2, 1), IV. (1, 1, 1, 1). Therefore, in the GC, it is not possible a coding option with only one codon ending with a pyrimidine (Py), while with the purine (Pu) is; or with two codons, of which one is ending with Py and the other with Pu; and also there is no possibility that all four of quartet-codons have to be coding for four different AAs.
204 2. Four stereochemical types of AAs
Regarding to the second alphabet of GC, the amino acid alphabet, it was shown that 20 AAs are / 3H' 454C )64 *)4' 1996), as it is here presented in Figure 1 and Tables 2.1, 2.2 and Table 3.
A 04 (09) 13 L .. .. A 04 13 L 09 ..... 21 S 05 08 T S 05 08 T 10 (09) 19 05 11 C 05 11 M C M 12 21 D 07 10 E D 07 10 E 08 30 (08) 38 11 N 08 11 Q N Q 23 28 K 15 17 R K 15 17 R
H 11 18 W H 11 18 W 25 (08) 33 25 ..... 33 F 14 15 Y F 14 15 Y Table 2.1. The atom number within 8 pairs of alanine stereochemical type of AAs. On the full line, as well as on the dotted one, there are 86 atoms; the differences 8 and 9 (9 - 8 = 1) express the minimum change relation among the amino acids (Swanson, 1984, p 191). The order follows from the atom number hierarchy. For details see in Remark 2.2. By that notice that within outer class of AAs (AL + FY + HW) there are [(4 + 33 = 37) + (13 + 25 = 38) = 75] of atoms; and within inner class [(10 + 38 = 37 + 11) + (19 + 30 = 38 + 11) = 75 + 22 = 97]. By this only one amino acid (G) belongs to the stereochemical type of glycine; also only one amino acid (P) belongs to the type of proline; the pair V-I belongs to the stereochemical type of valine; and, finally, to the stereochemical type of alanine belong all other, the 16 amino acids. The presented classification comes from the amino acid conformation states. However, the same four stereochemical types, two and two (G, A and V, P), following our idea, come also from the amino acid constitution structures, in the following manner. The side chain of glycine ( - H) comes from the shortest possible hydrogen chain (H - H), and none of the other 19 amino acids has a hydrogen chain of this kind. The side chain of alanine ( - CH3), or, in relation to glycine, ( - CH2 - H) follows from the shortest possible noncyclic hydrocarbon chain (CH4), and still 15 amino acids have the alanine M the same side chain in the form of (- CH2 - R). The side chain of valine (H3C- [-CH3) follows from the shortest possible cyclic hydrocarbon, from cyclopropane, with a permanent openness and with a linkage to the head of amino acid through only one vertex of cyclopropane triangle; still only one amino acid, isoleucine, belongs to this type with the side chain H3C- [-CH2-CH3. The proline type (only with proline) follows from the same
205 source (cyclopropane), but with a permanent non-openness and with a linkage to the head through two vertices of cyclopropane triangle.
G 01 .... (00) .... 01 G
V 10 (03) 13 I
V 10 (03) 13 I
P 08 .... (00) .... 08 P
Table 2.2. The atom number within four pairs of non-alanine type of AAs. The two lines have 32 atoms each. The number of amino acids must be doubled through the pairs G-G and P-P; the amino acid order comes from the atom number hierarchy. D / # ' O-P AAs, all from alanine /' OP # other three types (G, P, V-I) (Tabs 2.1 & 2.2 and Box 2.1).
Box 2.1. Molecule pairs hierarchy in relation to natural numbers series (The generating of the Table 2. 1) 1. Aliphatic AAs (simpler than aromatic) 1.1. Hydrocarbon AAs (start: 04 atoms); 1.2. First possible OH derivatives (start: 05 atoms); 1.3. Sulfur OH analog, i.e. SH derivatives (start: 05 atoms; S > O); 1.4. Carboxyl group derivatives (start: 07 atoms); 1.5. Amide derivatives (start: 08 atoms); 1.6. Amino derivatives (start: 15 atoms); ------2.2. Heteroatom derivatives (start: 11 atoms); 2.1. Aromatic hydrocarbon and its OH derivative (start: 15 atoms); 2. Aromatic AAs (more complex than aliphatic) [Remark `EO # # ` /O ` O' # G ' `- O e contact is mediated by a CH2 group (only within threonine that group is methyl-substituted: H M C M CH3).]
206 Viewing that valine type is arranged into the pairs (although only with one pair) we researched the alanine type following the same concept. The result was as it is shown in Table 2.1. There are eight pairs of alanine type, classified into two subclasses. The three amino acid pairs belong to the first (outer) subclass: 1. with hydrocarbon noncyclic aliphatic chain (A-L), 2. with aromatic chain, without hetero atom (F-Y) and 3. with aromatic chain, with hetero atom (H-W). The five other and different pairs belong to the second (inner) subclass, with the presence of oxygen, nitrogen or sulfur atoms within a noncyclic side chain, in the form of a functional group: 1. with hydroxy group (S-T), 2. with sulfur atom (C-M), 3. with carboxylic group (D-E), 4. with amide group (N-Q) and 5. with amino group (K-R). This classification into two subclasses is accompanied by specific balances of number of atoms within amino acids molecules (within their side chains).
Table 3. Atom Number within 12 doublets and 8 triplets of 24 Amino Acids. On the full line, as well as on the dotted one, there are 118 atoms; HP: hydropathy index (Kyte and Doolittle, 1982) on a number unnamed scale; CE: cloister energy (a form of the free thermodynamical energy) in kcal/mol (Swanson, 1984). For details see the text. [Remark 2.2. (a) In five pairs of AAs within inner subclass, in Table 2.1, there are 97 atoms, that means 11 atoms more than in a half of total atom number within whole system of eight pairs (172 : 2 = 86; 86 + 11 = 97). (b) Within three pairs in outer subclass there are 75 atoms, what means 11 atoms less in relation to the same half (86 M 11 = 75). The arithmetical pattern (75-86-97) we will also find in the system presented in Figure 4, which follows from Figure 3 and Table 6, but not for the atom number than for the nucleotide number within the codons. On the other hand, the half of half (Table 2.1) stands in the same relation to atom number within four contact AAs (86:2 = 43; 43 M 32 = 11). (c) The 97 atoms within inner
207 subclass in Table 2.1. plus 32 atoms within contact AAs in Table 2.2 equals 129 atoms; then, 129 atoms plus 75 (43 + 32) atoms within outer subclass equals, in total, 204 atoms within 20 amino acid molecules, in their side chains. Cf. this arithmetical pattern, 129-75-43-32, with the same one in Figure 10 and Remark 4.4. (d8 O able of P 3 > # 8' Table 4.] One must notice that the amino acid order placed on the left side in Table 2.1 reflects two things at the same time M the grading continuity of pairs and cycling: 1-2-3-2-1 (AL M HW M FY M AL ). But there is a chemical justification for the sequencing of the pairs M all times two in each step M as given on the right. The S in relation to A as its first possible OH derivative; NQ M KR as nitrogen AAs; HF M WY as aromatic AAs. If so, then there remains only one, albeit chemically "impossible" situation, but the system has no choice but to be so as it is (CD M ME). In some way unexpected, but also this different chemical classification is also accompanied by arithmetical regularities. First, there is a diagonal connection of pairs, as shown in Equations (2). The second arithmetical regularity follows from the four amino acid sequences, to which have been attached the contact amino acids in the order of molecule size; in such a new arrangement follow very # $ R
CD 12 + QR 28 = 40 AS 09 + WY 33 = 42 (2) ME 21 + NK 23 = 44 LT 21 + HF 25 = 46 [Remark 2.3. (a) The balance of atom number within the class of alanine stereochemical type: 40 + 46 = 42 + 44 = 86 (86 x 2 = 172; 172 + 32 = 204);2 (b8 # $ R algorithm (Section 4.3): (STLA-G 31), (DEMC-P 41), ... (51) ... , (KRQN-V 61), (FYWH-I 71); (LA-MC-QN-WH 81), (ST-DE-KR-FY 91)3 (Figure 9 & 10).]
If two separate systems, presented in Tables 2.1 and 2.2, are viewed as subsystems, we can then integrate them into a single system, as it is presented in Table 3, composed of doublets and triplets4. If even just one amino acid, in the system in Table 2.2, appears twice, then all others
2 The 32 atoms within four contact AAs: G, P, V, I; cf. the quartet 40-42-44-46 valid for 16 AAs of alanine stereochemical type with this one 48-50-52-54 valid for the whole amino acid system in Tab. 9. 3 Two missing members to the full Gaussian algorithm are "hidden" within the two pairs of contact amino acids (GV-11 and PI-21). Notice that these two pairs we can find in Table 2.2 and they also appear to be pairs in an enzyme determination (Figure 8). 4 O+ hat out of all doublet-triplet systems, this is the only one with two possible distinctions for doublets (i.e. six and six, and then, three and three doublets) and three possible distinctions for triplets (i.e. four and four, then two and two, and, finally' 8P 3 )64 < *)4' 4478
208 must be represented twice. Then, in the integrated system (Table 3) the ratio of the number of contact and non-contact AAs is 8:16 = 1:2, corresponding to the symmetry in the simplest case 3- ' 4548 ; ; 30) 454' )64' 2004b). On the other hand, the ratio of the differences of the number of atoms in the integrated system is 8:12 = 2:3. If we add to all those relationships still the relationship between the differences of atoms in the system of Table 2.1 (which is 8:9) we reach the Platonic-Pythagorean system of harmonic musical scale, where also i ` O # > whole molecules of canonical AAs (384) (Table A.2 in Appendix). Immediately it is obvious that in this new system (Table 3) the proline (P) as a heterocyclic molecule, has to go together with histidine and tryptophan. This leads to two triplets. Valine, which must be twice represented, from chemical reasons best fits in the set of serine and cysteine (S, C, V), and still once with alanine. However, with alanine also fit well glycine, and by this all possible triplets have been completed; eight of them, at the same time arranged in 12 doublets (pairs). As we can see the pairing process is realized through a full accordance with the parameters for the polarity, the hydropathy index, HP (Kyte and Doolittle, 1982)5 and the cloister energy, CE, as a form of thermodynamycal free energy (Swanson, 1984); in all but one cases (phenylalanine is an exception: although non-polar, it pairs up with polar tyrosine; for further exceptions of phenylalanine see Footnote 10 and Remark 4.3); also through a strict atom number balance (118/118 atoms in both zigzag lines). By this, the order of doublets and triplets is established with a strict atom number increasing from one to another next amino acid in correspondence with the involvement of the biosynthetic precursors one after another: first, the less complex and U 3 )64 < *)4' 4478
3. Four diversity types of AAs
Shcherbak (1994) pointed out that the number of nucleons within two classes of AAs (the four- codon and non-four-codon amino acids), appears to be equal with multiples of the number 037, a very specific and unique number. But among the arithmetical regularities valid for number 037 in the decimal numbering system, Shcherbak has shown that such a unique number is also present in the system of numbers which are analogs of 037 in some other numbering systems; their values for q (numbering system basis), going from one to another, differ by three units. Written together without specifying the numbering basis, these numbers-analogs (Shcherbaks' "Prime Quantums", PQ) are as follows: 13, 25, 37, 49, ... As we can see, the first digit belongs to the series of natural numbers, and the second to the series of odd integers.
5 It appears that amino acid arrangement within Genetic Code Table (GCT) stays in a strict balanced relation to the polarity, expressed through hydropathy (Table A.3 in Appendix).
209 Remark 3.1. The uniqueness 0f number 037 is because the multiplication through the module (q - 1) preserves all three digits: (1 x 037 = 037; 10 x 037 = 370; 19 x 037 = 703), (2 x 037 = 074; 11 x 037 = 407; 20 x 037 = 740) etc. Remark ? 0 )R / /= @In the close vicinity of the decimal system, for example, some number systems with the bases 4 (Quantum 134 = 7), 7 (257 = 19), 10 (37), 13 (4913 = 61), etc, have the same periodic features for three-digit numbers" (Shcherbak, 1994). Knowing this, it is easy to see the possibility of a specific presentation of a series of natural 6 numbers, in the form of a matrix scheme (n x 11q) (Tables 4 - 5) ; such a scheme (TMA) along # 0 )R ' ?: (N = 13, 25, 37, 49, 5B, ...). But as a noteworthy is the fact that in TMA specific and unique arithmetical regularities characterize not only number 37 but also its neighbors, numbers 26 and 48, each in its own way. 3.1. The uniqueness of the number 26 If the first diagonal neighbor of the number 26, the number 16 is added to the number 26 and its two followers (17 and 18) are successively added to the obtained result, we get the results as in Equations-solutions (3): 26 = 26 26 + 42 + 59 + 77 = Y 16 +17 + 18 = Z 26 + 16 = 42 Y = 204 Z = 51 (3) 42 + 17 = 59 Y/4 = 51 Z = Y/4 59 + 18 = 77
With three adding (16 +17 +18 = 51 = Z) we obtained three new results, and with the inclusion of the initial number 26 M four results. Their sum is 204 (26 + 42 + 59 + 77 = 204 = Y = 4Z), exactly four times greater than the sum of the three adding (16+17+18 = 51 = Z). But this connection of two equalities is a single and unique case in the entire system of numbers within Table 4, in other words within the set of natural numbers.7
6 J # O P 3 -8' > 11 in two digit position, as it is said in Remark 2.2 d. 7 Moreover, it appears that this is the zeroth case in the 4th column of Table 4 in the decimal numbering system, and also the zeroth c # 0 )R / (q = 4, 7, 10' ?' 7' G 8C in all other cases, discrepancies arise 3 E E )64' > 8.
210
... (-2) ...... -22 (-1) -21 -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 (0) -10 -09 -08 -07 -06 -05 -04 -03 -02 -01 00 (1) 01 02 03 04 05 06 07 08 09 10 11 (2) 12 13 14 15 16 17 18 19 20 21 22 (3) 23 24 25 26 27 28 29 30 31 32 33 (4) 34 35 36 37 38 39 40 41 42 43 44 (5) 45 46 47 48 49 50 51 52 53 54 55 (6) 56 57 58 59 60 5B 62 63 64 65 66 (7) 67 68 69 70 71 72 6D 74 75 76 77 (8) 78 79 80 81 82 83 84 7F 86 87 88 (9) 89 90 91 92 93 94 95 96 97 98 99 (A) A0 A1 A2 A3 A4 A5 A6 A7 A8 A9 AA (B) B1 B2 B3 B4 B5 B6 B7 B8 B9 BA BB Table 4. The Table of minimal adding in decimal numbering system. A specific arrangement of natural / ' > ' # 0 )R @ @ analogs (13, 25, 37, 49 ...) within the first diagonal. For details see the text. (1) 01. 02. 03. 04. 05. 06. 07. 08. 09. 0A. 0B. 0C. 0D. 0E. 0F. 10. 11 (2) 12. 13. 14. 15. 16. 17. 18. 19. 1A. 1B. 1C. 1D. 1E. 1F. 20. 21. 22 (3) 23. 24. 25. 26. 27. 28. 29. 2A. 2B. 2C. 2D. 2E. 2F. 30. 31. 32. 33 (4) 34. 35. 36. 37. 38. 39. 3A. 3B. 3C. 3D. 3E. 3F. 40. 41. 42. 43. 44 (5) 45. 46. 47. 48. 49. 4A. 4B. 4C. 4D. 4E. 4F. 50. 51. 52. 53. 54. 55 (6) 56. 57. 58. 59. 5A. 5B. 5C. 5D. 5E. 5F. 60. 61. 62. 63. 64. 65. 66 (7) 67. 68. 69. 6A. 6B. 6C. 6D. 6E. 6F. 70. 71. 72. 73. 74. 75. 76. 77 (8) 78. 79. 7A. 7B. 7C. 7D. 7E. 7F. 80. 81. 82. 83. 84. 85. 86. 87. 88 (9) 89. 8A. 8B. 8C. 8D. 8E. 8F. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99 (A) 9A. 9B. 9C. 9D. 9E. 9F. A0. A1. A2. A3. A4. A5. A6. A7. A8. A9. AA (B) AB. AC. AD. AE. AF. B0. B1. B2. B3. B4. B5. B6. B7. B8. B9. BA. BB Table 5. The Table of minimal adding in hexadecimal numbering system. A specific arrangement of U / ' > ' # 0 )R "simulation" analogs (13, 25, 37, 49 ...) within first diagonal. For details see the text.
211 The uniqueness of the number 26 is expressed not only through the difference 26 - 10 = 16, but also through the sum 26 + 10 = 36, where number 36, as the second diagonal neighbor of the number 26, appears to be the member of a unique pair 25-36; unique case in the entire system of numbers within Table 4, and that means within the set of natural numbers. Namely, the numbers 25 and 36 are neighbors in third column of Table 4 with a difference of 11 as in all other cases, in all columns. But their square roots, as integers, appear to be also neighbors, and that is the said uniqueness (Equations-solutions (4):
2 x1 + y1 = 36 = 6 (x1 = 26; y1 = 10) 2 x2 + y2 = 25 = 5 (x2 = 17; y2 = 08) (4) 2 x1 M y1 = 16 = 4 2 x2 M y2 = 09 = 3
3.2. The unity of two realities M chemical and mathematical Through a purely chemical analysis it is possible to find such an arrangement of protein amino acids that fully corresponds to the observed arithmetical regularities, related to the uniqueness of number 26. Namely, Figure 3 shows an arrangement of AAs (in the 4 x 5 system), with the number of atoms as in Equations (3). This arrangement follows from a classification and systematization of 20 protein AAs into four diversity types8, as it is shown in Figure 2: first row in Figure 3 comes from the vertical line of the circular model in Figure 2; fourth row from the horizontal line; second row from four middle points of circular model; third row make two left and two right AAs in relation to two middle points (M & D); finally, fifth row make two left and two right AAs in relation to two middle points (A & R). By this, each sequence in Figure 3 is arranged by the size of molecules, i.e. by the number of atoms in the side chains,9 going from left to right; and the order of sequences is determined by the size of the first amino acid molecule in the sequence.10 Only with such a precise and strictly regulated system can we get the desired result, the sequence 26-42-59-77, signifying the number of atoms in the four columns of AAs, in their side chains (cf. Table 4, Equations (3) and Figures 2 & 3). Bearing in mind that an arithmetical law, presented in Equation (3), is fully confirmed, it makes sense to set up a working hypothesis, related to an algebraic law, presented in Equations (4): it must be that the quantities, given in Equation (4), x1 = 26, y1 = 10 as well as x2 = 17, y2 = 08, in a way also have been contained in the genetic code. Table 6 is an obvious and direct evidence for this. The first diversity type of AAs (P, G) and corresponding 08 codons appear to be diagonally
8 More details on the four diversity types of AAs (2 + 4 + 6 + 8) as it is presented in linear model of Figure 2, about / # # ' 3 )64' > 8 9 The only exception is valine, which is understandable enough when we know that valine and isoleucine belong to the same stereochemical type, the valine type. That sequence should, therefore, be understood as follows: two AAs of the alanine stereochemical type (N, Q) are followed by two AAs of the valine stereochemical type (V, I). 10 The dilemma whether before N or P is resolved by the following pairs: N is followed by a smaller pair V(10) M I(13), while P by a larger one L(13) M K(15).
212 on the right within Genetic Code Table (GCT), designated in Table 6 in light tones; the second one (L, I, V, A) with 17 codons diagonally on the left (in dark tones); altogether, in both diversity types of chemically low-level-functions (possessing only carbon and hydrogen within side chain) there are 25 codons. The third diversity type of AAs and corresponding 10 codons follows, within GCT (Table 6), in the next order (light blue tones): in column "U" up and down, and in columns "A" and "G" only up. The fourth type, with 26 codons (the dark blue tones), in column "C" up and down, and in columns "A" and "G" only down; altogether, in two diversity types of chemically high-level- functions (except C and H, possessing still O, and/or N, and/or S)11 there are 36 codons.
G 01 S 05 Y 15 W 18 39 78 A 04 D 07 M 11 R 17 39 102 24 C 05 T 08 E 10 F 14 37 13
102 N 08 Q 11 V 10 I 13 42 89 P 08 H 11 L 13 K 15 47 26 42 59 77 16 17 18 (1 x 68) (2 x 68) Figure 3. A specific classification and systematization of amino acids which follow from four diversity types (Figure 2). In the shadow space there are 20 AAs with atom number in molecules side chains. Within first two and last two columns: 1 x 68 and 2 x 68 atoms, respectively. Within two inner and two outer columns: 102 ± 1 atoms. Regarding at the rows: there are 78 atoms within first two and 78 +11 = 89 within last two rows; within first half of the middle row 13, and within the second one 13 +11 atoms. Within two halves of shadow spaces (light and dark) there is also a specific balance: 102 ± 00 atoms. All amino acid sequences are of the growing series from the aspect of number of atoms; all but one, in which Q-11 precedes V-10, because different stereochemical types have been distinguished: NQ belongs to alanine but V-I to valine type.
11 The only one exception is phenylalanine (F) which possesses within side chain the benzene with only carbon and hydrogen atoms. But the aromatic function is more complex than aliphatic one within the first two types of diversity. It is interesting (as we said in last paragraph in Section 2) that phenylalanine is also an exception in the pairing process in the integrated system in Table 3: although non-polar, it pairs up with polar tyrosine. In addition, phenylalanine is also an exception in the determination by two classes of enzymes aminoacyl-tRNA-synthetases (see Remark 4.3.).
213 X # / $E 3 78 @@ @ #@ first glance seem to be arbitrary and optional. However, if such a classification is viewed from the aspect of key qualificatives M the first half of the GCT (of pyrimidine kind) is of lower rank, and the second half of a higher rank (of purine kind) M then we get here also to the very strict quantitative relations (Figure 4). The Figure 4, in fact, shows that the principle of self-similarity (through the alignment into 25/36 codons) achieved in a way. First, if a series of I-II-III-IV types is taken as splitting into two classes, then in the first class belong two types from odd positions, and in the second one two types from even positions. And then "enters" at the scene the validity of the following regularity: the number of low-ranking codons in two types at the odd positions, plus the number of higher-ranking codons in the two types at the even positions is the same as the number of all codons in the first two diversity types (25). The reverse is true for the reverse situation: the number of low-ranking codons in the two types placed at the even positions, plus the number of higher-ranking codons in two types placed at the odd positions is the same as the number of all codons in the second two diversity types, the third and fourth, just 36. [From Figure 4 it follows that this zigzag regularity is valid also for the number of nucleotides.]
a b U,C A,G U,G C,A I 04 04 25 12 12 12 12 II 17 00 31 20 31 20 20 10 19 11 III 03 07 36 IV 08 18 30 48 29 49
25 o/e 100 83 100 83 36 (93) (90) (91) (92)
U C A G U,A C,G 75 11 I 02 10 02 10 75 04 20 II 19 12 08 12 27 24 86 11 III 14 06 05 05 108 19 11 IV 10 20 29 19 39 39 97 183 11 86 97 108
183 + 9 = 192 (89) (94) 89-90-91-92-93-94 Figure 4. This Figure follows from Figure 3 and Table 6. The designations I-IV: Four diversity types of amino acids. In first quadrant there is the number of three-letter words, i.e. codons within GCT (in total 61), while in other three quadrants M the number of letters (nucleotides). The regularity valid for zigzag
214 line is given in the forth paragraphs of Section 3.2. (a) The first half of Table 6. (b) The second half of Table 6. In OP / ) ' ' # OP / # 3/ 8 - evident; also the changes for 01, 10 and 11. Besides the obvious, there are some "hidden" regularities as, for example, the redistribution of 75/108 letters through the doublets. Thus, in the first two types (I, II), which possess chemical properties of lower rank, we have: UC43 - AG32 = 11 and CA43 - CA32 = 11. (Cf. with the same pattern, 43-32 in Remark 2.2b and in Table A.4.1.) On the other hand, the doublets AG32 and CA32 appear to be in a unit change relation with UA31; then, the doublets UC43 and UG43 with CG44. Similarly is in doublets of two other types (III, IV), which possess chemical properties of higher rank: CA60 - UC50 = 10 and AG58 - UG 48 = 10. On the other hand, the doublets AG58 and UC50 appear to be in a zeroth change relation with the doublets UA58 and CG50. It goes without saying that here is a case of coherence between codons and AAs, and also between AAs and an arithmetical system (Equations (3) as well as codons and an algebraic system (Equations (4). That the presented coherence, except all the said, even more is stylized and sophisticated, shows the order of the AAs (related by codons) in GCT (Table 6) which is also followed by atom number balance as it is presented in Remark 3.3. [Remark 3.3. Two halves of amino acid molecules, connected through a base-acidic loop in position K-D12 within GCT: (1.F-14, 2.L-13, 3.I-13, 4.M11, 5.V-10, 6.Y-15, 7.H-11, 8.Q-11, 9.N-8, 10.K-15) + (1.S-5, 2.P-8, 3.T-8, 4.A-4, 5.C-5, 6.W-18, 7.R-17, 8.G-1, 9.E-10, 10.D-7) = (Odd/Even 56/65 = 76-11) + (Odd/ Even 56 - 11/76:2) = Odd/Even 102±1 as a balance M the tab on t . 3 : )64' >>98]
12 / I $ R between second and third row in Figure 9 (in relation to Remark 2.3b).
215
2nd letter 1st 3rd U C A G UUU UCU UAU UGU C U UUC F UCC UAC Y UGC C U uua UCA S UAA UGA A L CT CT uug UCG UAG UGG W G cuu ccu CAU CGU U cuc ccc CAC H CGC C C L cca P CAA CGA R A cua Q cug ccg CAG CGG G auu ACU AAU AGU U N S auc I ACC AAC AGC C A ACA T AAA AGA A aua K R AUG M ACG AAG AGG G guu gcu GAU ggu U guc gcc GAC D ggc C G V A GAA gga G A gua gca E gug gcg GAG ggg G
Table 6. The standard genetic code with new design. The design responds to the classification of protein AAs into four classes, correspondently with four diversity types. The first diversity type (GP): the 8 codons in small non-bolding letters; second type (ALVI), the 17 codons in small bolding letters; third type (CMFYWH), the 10 codons in large letters and light shadow tones; fourth type (STDENQKR): the 26 codons in large letters and dark shadow tones. The three codons which are cross out, are the "stop" codons. 3.3. The uniqueness of the number 48 Now we will demonstrate the uniqueness of number 48 within the system of numbers in Table 4 with one exercise: Find a number in this system, in series of odd or even numbers, which with its three followers gives the same result (204) which we obtained with number 26, as it is shown above. It is immediately obvious that this number is 48, as it is shown in Equation (5): 48+50+52+54 = 204 (5) It is also immediately obvious that this is the middle row in the system of numbers (in decimal numbering system) of TMA in Table 4. Thus, it is inevitable that every two rows at the same distance from the middle row (in analogous positions) yield twice the value of number 204 [Example: (37+39+41+43) + (59+61+63+65) = 2 x 204).] In addition, in relation to the two ends, and the two central numbers of the sequence 48-50-52-54, there is a symmetrical and cross- / 3 E9 )64' > 8
216 Through an exact chemical analysis it is also possible to find such an arrangement of protein amino acids that fully corresponds to the observed arithmetical regularities, related to the uniqueness of number 48. By this we start from an elegant hydrogen atom number determined system given by V. Sukhodolets (1985) (Table 7 in relation to Table 8). The system in Table 7 consists of two subsystems; the first subsystem: the left (small) subsystem consisting of four AAs (G, N, Q, W) in the form of four singlet sequences [(G), (N), (Q), (W)]; the second subsystem: the right (large) subsystem of 16 AAs, in the form of four quaternary sequences [(ASDC), (PTEH), (VFMY), (RKIL)]; in the subsystem with 4 AAs, the outer are G and W; the inner ones N and Q; in the subsystem with 16 AAs, alanine (A) and cysteine (C) are the outer; serine (S) and aspartic acid (D) the inner ones, and so on. After this analysis we can generate the left subsystem of Table 9: within the first column ("out") are the outer, and within the second one ("in") the inner AAs, given in the same order as in 0) R / : Now the next question makes sense: is it possible to sort AAs in both columns in accordance with their fundamental chemical nature? The answer is affirmative, as it is demonstrated in the right subsystem of Table 9. (For details see Section 4.2.1. in )64' > 8 The number of H atoms (in brackets) and nucleons G (01) 01 A (03) 15 S (03) 31 D (03) 59 C (03) 47 (13) 153 N (04) 58 P (05) 41 T (05) 45 E (05) 73 H (05) 81 (24) 298 (59/58) Q (06) 72 V (07) 43 F (07) 91 M (07) 75 Y (07)107 (34) 388 569/686 W (08)130 R (10) 100 K (10) 72 I (09) 57 L (09) 57 (46) 416 569 as neutron number and 686 as proton number! 569 M 59 = 627 M 117 686 M 58 = 628
Table 7. 0) R / 9 U 6 0) R ' # minimal modification (Sukhodolets, 1985): the system of 4 x 5 AAs. The shadow space: AAs with even number of hydrogen atoms (4, 6, 8, 10); the non-shadow space: AAs with odd number of hydrogen atoms (1, 3, 7, 9, 11). In brackets: number of hydrogen atoms and out of brackets the number of nucleons. Nucleon number through a specific "simulation": 569 within two outer rows, as the number of neutrons, 569, in all 20 AAs M within their side chains; and 686 nucleons within two inner rows, as the number of protons, 686, in all 20 AAs M within their side chains. [The "simulation" as a holistic information within a part abo # O 3 )64' > 8B M > 674 neutrons as well as 569 non-hydrogen protons. Within 20 side chains of amino acid molecules there are 117 hydrogen protons, what means 117 hydrogen atoms at the same time (117 = 59 + 58).
217
G P A L V I C M F Y W H R K Q N E D T S
01 08 04 13 10 13 05 11 14 15 18 11 17 15 11 08 10 07 08 05
9+18=27 40+36 =76 74 + 54 = 128 81+ 72 = 153 (27+153 = 180) (76+128 =204)
Table 8. Four types of diversity of protein amino acids. The relationships between four diversity types of protein amino acids (2+4+6+8). In second row there is the number of atoms within side chains of amino acids. The calculations: within 10 AAs of two inner types there are 180 atoms, just as within 20 amino acid "heads", i.e. 20 amino acid functional groups (20 x 9 = 180). On the other hand, within 10 AAs of two outer types there are 204 atoms, just as within 20 amino acid side chains. This specific "simulation" is analogue to the "simulation", valid for the number of protons and neutrons in Table 7.
out in out in G (01) N (08) G (01) S (05) W (18) Q (11) A (04) T (08) A (04) S (05) L (13) I (13) C (05) D (07) V (10) D (07) P (08) T (08) P (08) E (10) H (11) E (10) R (17) K (15) V (10) F (14) Y (15) F (14) Y (15) M (11) W (18) Q (11) R (17) K (15) H (11) N (08) L (13) I (13) C (05) M(11)
O 40 50 48 50 E 62 52 54 52 102 102 102 102
Table 9. Atom number within 20 side chains of AAs as a quartet of even numbers. The outer/inner amino # # 0) R 3 :8 G = :C on the right: the chemical order of AAs as it is explained in Section 3.3.
218 4. Harmonic structures
A proof more that genetic code is a coherent system is the fact that there are some harmonic # ' # #) 3 )64' 44:' 445' 2004a, 2006, 2009, 2011). By this the term (and concept) "harmonic" is not used as a metaphor but as concrete mathematical expression, such as golden mean, harmonic mean, arithmetic mean etc.
4.1. Determination by Golden mean
D ' # # 3 )64' 4458 ) $E ' m as in Table 6, developed into a binary-code tree (0-63), exactly in the order specified here: first, codon octets with both pyrimidines, and then with both purines (F, L, L), (S, P), (I, M, V), (T, A), (Y, CT, H, Q), (C, CT, W, R), (N, K, D, E), (S, R, G). Such a tree appears to be in a strict correspondence to the Farey tree, including the correspondence between the route 0f most change (at each step turns ; = > > > >8 O P # D ' numerators and denominators: 1/2, 2/3, 3/5, 5/8, 8/13, 13/21, 21/34. (Notice that Fibonacci numbers are in a strict relation to golden mean.) By this, in the case of six-bit binary tree and the adequate Farey tree last fraction point (21/34) stays in relation to number 0101010 (42) which number represents the harmonic mean of the interval 0-63 and its half. In this point stays the codon with most diversity M the stop codon UGA; its neighbor is the codon UGG which codes for the most complex amino acid, tryptophan(W), only and one with two aromatic rings. As we see, the sense is clear: to balance the great diversity with maximum of harmony! (Cf. Fig. 1 and D )64' 4458 Bearing in mind above presented harmonic relations it makes sense to determine (calculate!) the golden mean for interval 0-7?' ; ' n (n = 0, 1, 2, 3, ... , 48 30) ' 454' )64' >>98' > U/ positions of the golden mean.13 They are G, Q, T, P, S, L, F with 60 atoms. Across from them must be seven of their chemically pairing "partners" as the complements: V, N, M, I, C, A, Y with 60 + (1 x 6) atoms; and the six remaining AAs, the three pairs of the non-complements: D- E, K-R, H-W with [60 + (1 x 6)] + (2 x 6) atoms; for them it makes sense that aliphatic AAs come first (D-E/K-R), followed by aromatic ones (H-W) (Figure 5).
13 It is self-evident that only at the six-bit binary tree the determination with the generalized golden mean is going through module 9, that is through the decimal numbering system. On the other hand, because two golden mean values are calculated from the quadratic equation by square root of the number five (which represents half of the decimal scale of 10 digits) it is reason why decimal numbering system is - the only "golden" numbering system.
219
G 01 10 V Q 11 08 N T 08 11 M P 08 13 I S 05 05 C L 13 04 A F 14 15 Y ------ D 07 10 E K 15 17 R H 11 18 W
1021 Figure 5. Atom number balance directed by Golden mean on the binary-code tree (presented in Figure 1 )64' 4458 D Q R ; the arrangement- ordering follows from the ordering given in Table 10. On the right are their complements; below are three amino acid pairs as non-complements. The atom number balance: On the zigzag lines there are 102±1 of atoms. Within three classes of AAs there are 60, 66 and 78 atoms (with the differences for 1 x 6, 2 x 6 and 3 x 6), respectively.
[Remark 4.1. The differences between three groups of amino acids, through the number of atoms (60, 66, 78), are 1 x 6, 2 x 6 and 3 x 6 respectively.] In next step it makes sense the amino acid order, presented in Figure 5, to be rearranged by the size of the molecules: first glycine with one atom in the side chain, then serine with 5, threonine with 8 atoms, follows proline (the pair T-08/M-11 before the pair of P-08/I-13) etc. If this new I O P # 5> ' # / system that we called CIPS (Cyclic Invariant Periodic System) (Figure 6).14 [Remark 4.2. The presented order of codons in Table 6 is based on the codon octets, but for a better viewing how the system, presented in Figure 6, follows from GCT (Table 6), it is also necessarily to have an order based on the codon quartets, such as it is presented in Table 11 (Négadi, 2009). Also to have a very symmetric form of GCT, valid for mitochondrial genetic code (Dragovich & Dragovich, 2010.] The further analysis shows that the order of five amino acid classes makes an adequate chemical sense: on the middle position (the position "1") came non-contact, the chalcogene AAs (S, T & C, M); follow (on the position "2") contact AAs G-P and V-I; then came (on the position "3")
14 Cyclicity and periodicity through the positions of two and two amino acids M up/down M in relation to middle chalcogene AAs, on the position "3".
220 next class of non-contact AAs, two very polar, double acidic AAs with two their amide derivatives (D, E & N, Q); follow (on the position "4") the source aliphatic AAs of the alanine stereochemical type (A-L and K-R), two of which are amine derivatives (K-R), with a lower degree of polarity (nitrogen is less polar then oxygen!); finally came four aromatic AAs (F,Y & H, W) on the position "5".
5 073 F 14 15 Y 079 4 235 L 13 04 A 172 3 087 Q 11 08 N 085 2 160 P 08 13 I 121 1 168 T 08 11 M 043 1 243 S 05 05 C 081 2 184 G 01 10 V 168 3 087 D 07 10 E 093 4 091 K 15 17 R 265 5 081 H 11 18 W 044
Figure 6. The Cyclic Invariant Periodic System (CIPS) of canonical AAs. At the outer side, left and right, it is designated the number of atoms within coding codons; more exactly, in the Py-Pu bases (U = 12, C = 13, A = 15 and G = 16); at the inner side M the atom number within amino acid side chains. In the middle position there are chalcogene AAs (S, T & C, M); follow - U `/O - the AAs of non-alaninic stereochemical types (G, P & V, I), then two double acidic AAs with two their amide derivatives (D, E & N, Q), the two original aliphatic AAs with two amine derivatives (A, L & K, R); and, finely, four aromatic AAs (F,Y & H, W) M two up and two down. The said five classes belong to two superclasses: primary superclass in light areas and secondary superclass in dark areas. Notice, that each amino acid EJH0 / 3 )64' >>48
From pure chemical reasons it makes sense to say that two classes (light tones in Figure 6) belong to a primary superclass, with original aliphatic AAs (and/or derivatives of lower level), whereas the three remaining classes (dark tones) to a secondary superclass, with the derivatives of a higher level.
221
0 1 2 3 4 5-7 8 9 G Q T P S L L-F F 63 39-38 25-24 15-14 10-09 06-02 02-01 01-00 63 38.94 24.06 14.87 9.19 5.68 M 2.17 1.34 0.83
Table 10. The determination of the amino acid positions through the golden mean. The distribution of AAs by the generalized Golden mean (through power values) within the sequence 0M63 on the binary- 3 D )64' 4458 D #= $ # # Q/R 9. Second row: amino acids in the positions marked in third row, taken from the binary-code tree. Fourth row: the values of the Golden mean powers within the interval 0M63. The calculations: 0.618033 x 63 = 38.94; 0.618033 x 0.618033 x 63 = 24.06 etc.
U-U-X U-C-X C-U-X C-C-X U-A-X U-G-X C-A-X C-G-X A-U-X A-C-X G-U-X G-C-X A-A-X A-G-X G-A-X G-G-X
Table 11. A model for codon distribution in Genetic Code Table. The codon order in Genetic Code Table, based on the codon quartets in first and second half of the Table, as in Table 1 in (Négadi, 2009), here simplified and generalized.
4.2. Golden mean M enzyme determination The splitting into two above presented superclasses exists in a strict correspondence to the splitting into two enzyme-directed classes (amino acids handled by class I and by class II of aminoacyl-tRNA synthetases), as it is shown in Figure 7 and Figure 8, in relation to Box 4.1. [Remark 4.3. There are two classes of aminoacyl-tRNA synthetases: Class I has two highly conserved sequence motifs. It aminoacylates at the 2'-OH of an adenosine nucleotide, and is usually monomeric or dimeric (one or two subunits, respectively). Class II has three highly conserved sequence motifs. It aminoacylates at the 3'-OH of the same adenosine, and is usually dimeric or tetrameric (two or four subunits, respectively). Although phenylalanine-tRNA synthetase is class II, it aminoacylates at the 2'- OH. The amino acids are attached to the hydroxyl (-OH) group of the adenosine via the carboxyl (- COOH) group.] From the correspondence between two superclasses, two classes and four subclasses as it is presented in Figure 7, can follow a prediction (Prediction 1) for further researches: the demonstrated crossing of two superclasses, two classes and four subclasses must be reflected, in
222 some way, in the protein structures and functions. [Additional texts about amino acid handling / # ;/ = 3 )64' 44:' >>48B
Box 4.1. Molecule pairs hierarchy in relation to natural numbers series (going from Tables 2. 1 & 2.2 to Figure 8)15
1. Vertical pairs in SubS1 (Tab. 2.1) and SubS2 (Tab. 2.2)
1.1. Glycine as the simplest (G-V) (start: 01 atom); 1.2. Serine as next (S-C) (start: 05 atoms); 1.3. Threonine as next (T-M) (start: 08 atoms); 1.4. Proline as next (P-I) (start: 08 atoms; M < I); 2. Horizontal pairs in SubS1 (Tab. 2.1) and SubS2 (Tab. 2.2) 2.1. Hydrocarbons (A-L) (start: 04 atoms); 2.2. Carboxyl group derivatives (D-E) (start: 07 atoms); 2.3. Amide derivatives (N-Q) (start: 08 atoms); 2.4. Amino derivatives (K-R) (start: 15 atoms); ------3.2. Heteroatom derivatives (H-W) (start: 11 atoms); 3.1. Aromatic hydrocarbon and its OH derivative (F-Y) (start: 15 atoms); 3. Aromatic AAs (more complex than aliphatic)
09 G P (2) 23 V I 28 53 81 19 A K (4) 30 L R
13 S T (1) 16 C M 53 15 D N (3) 21 E Q 70 123 25 F H (5) 33 Y W 81 123 204
Figure 7. The amino acid arrang = O P OP O P OP 0' 35 8 D 5 3 JJ' # smaller molecules within the pairs); and on the right there are AAs (123 atoms) from the right side of
15 For details about first subsystem, presented in Table 2.1 (SubS1) and second subsystem, presented in Table 2.2 3008 # O P / D 5' = 3 )64' 4458
223 Figure 8 (class I, with larger molecules within the pairs). At the same time very up there are AAs from / 35 8' & 3'%' .' J8 O P 3$' H' L' 8 (hydrogen and nitrogen are less polar then oxygen!); in the other hand, except aromatic and sulfur AAs, # / 3 # # ? 8' ' OP
G 01 10 V S 05 14 26 05 C T 08 11 M
P 08 13 I 26 A 0412 13 L
D 07 10 E N 08 30 38 11 Q K 15 17 R
H 11 18 W 25 33 F 14 15 Y 81 123
(102 ± 1)
Figure 8. Atom number balance directed by two classes of enzymes aminoacyl -tRNA synthetases. This system follows from the systems given in Figure 5 in relation to Tables 2.1 and 2.2 (cf. Box 4.1). As in the system given in Figure 5, on the zigzag lines there is a balance of 102±1 of atoms. Class II handles the smaller amino acids within the pairs (on the left), whereas the larger (on the right) are handled by class I aminoacyl-tRNA synthetases. R
There are several anecdotes about Gauss as a little boy and his understanding of mathematics at his earliest age. One of them say that the teacher gave an exercise to a class of young pupils (only nine years of age) to add all the numbers from 1 to 100, thinking he would have enough time to get on with some other things, while the pupils did the very long addition. However, the teacher was surprised when one pupil produced the correct answer in less than 3 minutes. The answer was 5050. Asked how calculated it, the young student explained that he added the first
224 and last numbers (1 + 100 = 101), then the following and the first preceding one (2 + 99 = 101) and so on. Since there are 50 such pairs, he multiplied 50 x 101 and obtained the requested result. The young pupil was named Gauss, and the algorithm he used to get the answer has become known as Gauss' Algorithm. !" #$ "!$%%#& $ '$ "$ "(!! "() &( $# #)"#* "! # +)! # !"( , + &!! #* +)! "!$ ) )#$ %# %!" #*! "%#$ "() -./"# 0 !"# -12
And now the reader is asked to look at the first row in Equation 6. We see that the particularly designated quantities (bold) are the same as the number of atoms within molecule side chains of AAs # O # P ' # Remark 2.3b and Figure 9 in relation to Figure 10. [Remark 4.4. Notice that the pattern (129-75-43-32) of atoms is the same as in Remark 2.2c. Arithmetical regularities background of this pattern as well as of the pattern 129 = 61 + 68 one can find in Table A.4.1 and A.4.2, in Appendix.]
Figure 9. $ R C 38 acids has been derived from r ' / # # O P J # #/ OP ' / 3 OP O-P o acids see in Remark 2.1.). Atom number (in amino acid side chains) in the rows and columns generated in this way corresponds, one ' I # $ R 101 (Equations-solutions (6). Dark tones: Class I of amino acids handled by class I of enzymes aminoacyl-tRNA synthetases; light tones: Class II. Going from (a) to (b) it is obviously that chemically
225 O) P / >' ' / (/ ; O P aromatics; by one step in chalcogenic AAs (M, C in relation to S, T) and carboxylic (carboxylic AAs D & E in relation with their amides N & Q); by two steps in source aliphatic AAs: A, L & K, R. The distribution in (b) is the same as in 38 OP properties of molecules, so there is no more taking off. The class of contact AAs has been added to the beginning of columns, instead of to the beginning of rows. Shading is the same as in (a). In Figure 10 are given formulae of 20 canonical AAs in order as in Figure 9. From those formulae, i.e. from the chemical structures it is easy to see that the four contact AAs (GPVI), with 32 atoms within their side chains, can be classified together with five other AAs (ALSDF), possessing 43 atoms, into the class of invariant AAs (light tones). The remaining 11 AAs constitute a class of variant AAs with a total of 129 atoms in their side chains. Furthermore it makes sense to break this class into two subclasses: the subclass of less variant AAs (TEMCQNY) with 68 atoms and the subclass of more variant AAs (KRWH) with 61 atoms.
Figure 10. The structure of amino acid molecules. The simplest amino acid is glycine (G) whose side chain is only one atom of hydrogen. It is followed by alanine (A) whose side chain is only one CH3 group, which is the smallest hydrocarbon group. There are total of 16 amino acids of alaninic stereochemical
226 / 3O-P 8 # E[2 group, each between th O /P O P glycine type contains glycine (G) only; valine type contains valine and isoleucine (V, I). The last stereochemical type is proline type with proline (P) which represents the inversion of valine in the sense
OP of three CH2 O P / ' / # one but with two CH2 3H' 454C )64 < *)4' 4478 . A3$' H' %' J83? atoms)] & [(A, L, S, D, F)(43 atoms)]: invariant AAs; most dark tones [(K, R, W, H)(61 atoms]: most variant AAs; less dark tones [(T, E, M, C, Q, N, Y)(68 atoms)]: less variant AAs. Notice a further possibility for splitting: 1. two aliphatic AAs [(K, R)(32 atoms)], 2. two aromatic AAs [(W, H) (29 atoms)], 3. chalcogene AAs plus aromatic hydroxide derivative [(MC, TY)(39 atoms), and 4. dicarboxilic amino acid plus two amides [(E, QN)(29 atoms) (Cf. Section 4.3.). A specific calculation: less variant AAs, all aliphatic but one (Y), plus two more variant (aromatic: W, H), equals 68 + 29 = 97 atoms; invariant AAs, all aliphatic but one (F) plus two more variant (aliphatic: K, R), equals 75 + 32 = 107 atoms; if so, then: 107 M 97 = 10; 107 + 97 = 204. [Additional calculations: 32 + 43 = 75; 61+68 = 129 (cf. Remark 2.2).]
Now the question what is the meaning of the concepts (notions) of "invariant" or "variant". It is easy to understand why, for example, the glycine and alanine are invariant molecules; a molecule (G) with the smallest possible non-hydrocarbon side chain, and a molecule (A) with the smallest possible hydrocarbon side chain. But, it is harder to understand why valine with a propyl group as its side chain is also invariant if there is (in the nature) simpler ethyl group. The explanation is contained in Figure 1. Here we see the sense, meaning and meaningfulness (logic?) of choice: after the first possible case of non-hydrocarbonicity come possible cases of hydrocarbonicity; the first possible case of openness, then of half-cyclicity, cyclicity and branching (hence the meaning of four stereochemical types of AAs). In such a choice it has to be so as it is (and so must be anywhere in the universe). If, however, could be possible second, third etc. cases, then the invariant AAs outlined here would not be invariant. Hence it makes sense to talk not only about more or less variant, but also of more or less invariant (Figure 11). Figure 11 shows that the splitting of AAs into two classes M variant and invariant M is not only a / 3 $ R 816, or just a matter of chemical structure of molecules, but also the matter of amino acid functions (polarity). In O P 3 invariant) there are six AAs, two polar (KR) and four semi-polar (GP and HW).17 In column O P 3 8 9 ' -polar. If, however, do not look at the number of molecules than at the number of atomic associations then they are seven polar and six non-polar. [The AAs leucine (L) and isoleucine (I) are one and the
16 The matter of formality is also the atom number pattern: (60 + 10), 66, (78 M 10) in relation to the pattern which follows from the golden mean determination: 60, 66, 78 as it is presented in Remark 5.1. 17 From the Table 3 we se that lysine (K) and arginine (R) are polar in both parameters: hydropathy and cloister energy. On the other hand glycine (G) and triptophan (W) are polar in hydtropathy and non-polar in cloister energy. The histidine (H) is polar in hydropathy, but neutral in cloister energy. The P is special case: it is polar from the aspect of hydropathy, and non-polar from t I 3M ' 477C LR ) ' 4:6C )64' >>9).
227 same atom association through the structural isomery.] Altogether, the amino acid molecule pattern expressed in Figure 11 appears to be 4-9-6, what means: 4 semi-polar, 9 polar and 6 non- polar.18 [About relations between polarity of AAs and their positions within GCT see Table A.3 in Appendix] more less G P V I A L F 66
S D invariant
H W T E Y K R N Q 68 variant C M
60 + 10 68 = 78 - 10 Figure 11. The variability of AAs with respect to their polarity; explanation in the text (cf. two last paragraphs in Section 4.3). From the showed distinctions into invariant and variant AAs can follow a prediction (Prediction 2) and from distinctions into semi-polar, polar and non-polar AAs also a prediction (Prediction 3) both for further researches: the demonstrated distinctions have to be reflected, in some way, in the protein structures and functions in healthy as well as sick states. 4.4. A specific harmonic structure From the system presented in Figure 8 follows a new very specific harmonic structure within a system of 5 x 4 of AAs as i 3 )64' >>98 The five rows within the system start with one polar charged amino acid each, making first column, consisting from five polar charged AAs (D, R, K, H, E). In Figure 8 the molecule pair D-E make the only two charged acidic molecules (dicarboxilic AAs). In their neighborhood are two their amide derivatives (N-Q). The following six molecules, in a strict order, appear to be two triplets; the first (three basic molecules) RKH and the second one (three neither acidic nor basic molecules), the triplet FYW. Actually, the first triplet ends with the sole aromatic amino acid, which is basic charged (H), and continue with three aromatic
18 As a noteworthy is the fact that the pattern 4-9-6 corresponds to the third perfect number 496 (cf. Table A.4.1 and 4.2 in Appendix).
228 non-electrified; and then: between the first two amino acids (D-E) comes the first triplet, and between the second two (N-Q) comes the second triplet. Thus, in such a manner is generated the left half of the system in Table 12; then comes the right half taking two sequences in a vice versa position from the upper part of the system in Figure 8. a b c d M D N A L 189 189 221 221+3 485.49 485 R F P I 289 289 341 341+0 585.70 586 K Y T M 299 299 351 351+2 595.71 596 H W S C 289 289 331 331+1 585.64 586 E Q G V 189 189 221 221+3 485.50 485 1255 1255 1465 1465+9 2738.04 2 (37 x 37) Table 12. # # O P O P I # O P " < T O P ' L < [ basic. Four choices after four types of isotopes: (a) The number of nucleons within 20 AAs side chains, calculated from the first, the lightest nuclide (H-1, C-12, N-14, O-16, S-32). (b) The number of nucleons within 20 AAs side chains, calculated from the nuclide with the most abundance in the nature [the same patterns as in (a): H-1, C-12, N-14, O-16, S-32; at heavier nuclides of other bioelements the data by (a) and (b) are not the same]. (c) The number of nucleons within 20 AAs side chains, calculated from the nuclide with the less abundance in the nature (H-2, C-13, N-15, O-17, S-36); (d) The number of nucleons within 20 AAs side chains, calculated from the last, the heaviest nuclide (H-2, C-13, N-15, O-18, S-36). (M) The AAs molecule mass. Notice that (d) is greater from (c) for exactly one modular cycle (in module 9) and that total molecules mass is equal to 2 (37 x 37). Notice also that molecule mass within five rows is realized through the same logic-patterns of notations as the first nuclide, i.e. isotope 3 )64' >>98
Within the first two columns in Table 12 [(a) and (b)] is given the number of nucleons within twenty amino acid molecules (side chains), calculated after the first i.e. the lightest nuclides (a), also the nuclides with the maximal abundance (b)19 (H = 1, C = 12, N = 14, O = 16, S = 32). In third column (c) is the number of nucleons, calculated according to the nuclides with lowest abundance in nature (H = 2, C = 13, N = 15, O = 17, S = 36); fourth column (d): the number of nucleons, calculated according to the latest, i.e. heaviest nuclides (H = 2, C = 13, N = 15, O = 18, S = 36); finely, in last column (M) is the calculated molecular mass.
19 OP OP are the same because the matter is about one and the same nuclide: the first nuclide (isotope) appears to be with the maximal abundance.
229 It should be noted that for nucleon number (within all nuclides) as well as for molecule mass, two principles are valid: the principle of continuity and unit change principle (Equations (7-9). That means that other possible balances of nucleon number as well as molecule mass, in accordance with a-b-c-b-a pattern, can not save neither the neighborhood of pair-members nor the validity of these two principles. By this, in the system in Table 12 the pairs are as in upper row of Equation-solution (10). In the bottom row there are the pairs as in the linear arrangement of the system consisting of four diversity types, shown in Figure 2. 189 M 100 M 289 M 10 M 299 M 10 M 289 M 100 M 189 (7) 221 M 110 M 331 M 10 M 341 M 10 M 351 (8) 485 M 101 M 586 M 10 M 596 M 101 M 586 M 101 M 485 (9)
(GV, PI); (SC, TM, AL, DE [DN, EQ], NQ, KR, HW, FY) (GV, PI), (AL, CM, FY / WH, RK, QN, ED, TS) (10) As we see from Table 12 the number of nucleons within 20 side chains of 20 amino acid molecules is 1255. Adding to this result the number of nucleons within amino acid functional group, 20 times (20 x 74 = 1480), we get a total of 2735; that means for three units less than molecule mass (2738 M 2735 = 3). But these three units enabled that the molecular mass appears / 0 )R )/ ' ?:' / I / specific. More than all that, despite all the other isotopes, all except the first, participate with a minimum percentage, the balance of the total number of nucleons in these isotopes is complete (columns c and d), as if they participate in the full amount.20 5. Concluding remark
The presented coherent and harmonic structures support the hypothesis of Sukhodolets (1985), as # # 3 )64' >>98' # complete, consisting of four amino bases and twenty amino acids. It remains, however, an open question of relationships between the standard genetic code (which we analyze here) and the so- called deviant codes. For example, the Table of mammalian mitochondrial code is more symmetrical than the Table of the standard one, because it does not possess asymmetric degeneracy (3-1), or (2-1-1), but only symmetrical (4) and (2-2). However, it fits only with the arithmetical system, given in Equation-solutions (3), but not with the algebraic, given in Equation-solutions (4). From this follows a new question: what is older - and who is here the chicken, and who the egg - the mitochondrial code which is determined with only one, or the standard one, determined with both mathematical systems? It is our hope that the answer to this question will be given in the future researches.
20 OM ' )# #/ ' / Q H R # ; P 3 )64' >>98
230 Appendix: Some source harmonic structures HR / # ' # I ' and second one with the quotient 3, both in relations to natural numbers series, their squares and the sum (first to last and the last column) of both. Notice that from two middle rows, 4th and 5th, the 4th appears to be in relation to genetic code entities. For example, the 4 & 16 as four contact and 16 noncontact AAs or 4 x 16 codons within GCT; the 81 as 61 amino acid codons plus 20 AAs; the 64 as 64 codons; the 20 as 20 AAs in a 4 x 5 arrangement. HR # # # 3 columns in Table A.1) follows a Pythagorean musicale tone scale. The order of four blocks as follows: 1. the formulas for harmonic and arithmetical means; 2. two progressions extended with harmonic and arithmetical means between two members in all steps, then combined; 3. first two members of progression with the quotient 2, the 1-2 interval, extended with harmonic and arithmetical means and then with additional two members through the rule given in row below in order to become a whole musicale tone scale (the quotient of all two neighbor numbers is 9/8, corresponding to a whole tone, except in two cases where it is 256/243, corresponding to a half- tone; 4. the reducing of fractions in the previous interval 1-2 in a common denominator, the number 384, wich number we can also find as a total number of atoms within 20 amino acid molecules. The Table A.3 shows that strictly arranged positions of AAs within GCT appear to be in a strong correspondence with the polarity of amino acid molecule and its size (atom number) through the validity of the principle of minimum change as follows, in the designations: (n) non-polar, (p) ' 38 ' 38 3 )64' >>>8= (n) 4V+1M+3I+4A+2L+4L+2F+2C = 22 molecules 40+ 11+39+16+26 + 52 +28+10 = 222 atoms (420) (11)
(o) 4V+1M+3I+4A+2Y+4R+1W+2C = 21 molecules 40+ 11+39+16+ 30 + 68 +18+ 10 = 232 atoms (421) (12)
(p) 4G+2K+2N+4P+2Y+4R+1W+2E+2D+4T+2R+2S+2Q+2H+4S = 39 04+30+16+32+30 + 68+ 18+ 20 +14+ 32+ 34+10+22+22 + 20 = 372 (723) (13) (i) 4G+2K+2N+4P+2L+4L+2F+2E+2D+4T+2R+2S+2Q+2H+4S = 40 04+30+16+32+26+52 + 28+20+ 14+32+ 34 +10+22+ 22+ 20 = 362 (722) (14)
From the above presented equations follows that in 22 molecules of non-polar AAs there are 222 of atoms within their side chains, and 420 of atoms within the whole molecules; in outer space, however: 01 molecule less, 10 of atoms more within side chains and 01 atom more within the whole molecules. On the other hand, in 39 molecules of polar AAs there are 372 of atoms within
231 their side chains and 723 of atoms within the whole molecules; in inner space, however: 01 molecule more, 10 of atoms less within side chains and 01 atom less within the whole molecules. The Table A.4.1. shows the changes through the application of the two principles (of minimum change and of the continuity), starting from the first possible two-digit number in decimal numbering system (from the number 10 in column a1 in correspondence to the first perfect number, the number 6, in column d). In this Table one can find the background of arithmetical patterns (32+43 = 75) and (61+ 68 = 129), given in Remark 2.2 and Figure 10.
The Table A.4.2. shows that all relations in Table A.4.1, ultimately are determined by the third perfect number, the number 496. x1 x2=y 2X 3X xy=x3 x+y=z z 0 0 1 1 0 0 0 x 1 1 1 2 3 1 2 1 x 2 2 4 4 9 8 6 2 x 3 3 9 8 27 27 12 3 x 4 4 16 16 81 64 20 4 x 5 5 25 32 243 125 30 5 x 6 6 36 64 729 216 42 6 x 7 7 49 128 2187 343 56 7 x 8 8 64 256 6561 512 72 8 x 9 9 81 512 19683 729 90 9 x 10 Table A.1 # HR
* * *
232 Harmonic mean (h) Arithmetic mean (m) 2ab a b h = m = a b 2
4 3 8 16 1, , , 2, ,3 , 4, , 6 , 8 3 2 3 3
3 9 27 1, , 2 , 3, , 6 , 9, ,18, 27 2 2 2
4 3 8 9 16 27 1, | , 2, |3 , 4| , | 6 , 8|9, ,18, 27 3 2 3 2 3 2
1, 9 , 81 4 | 3 , 27 , 243, 2 8 64 3 2 16 128
(32 & 34 /23 & 26) | (32+1 & 34+1 / 23+1 & 26+1)
384 432 486 512 576 648 729 768 48 54 26 64 72 81 39 (384) Table A.2 # HR # H/
,+!# &(# #$ % ## ##34+$#%!#
233
e a1 b a2 cd 10 20 10 + B9 = 129 06 + 123 = 129 160 31 21 31 21 + A8 = 129 17 + 112 = 129
(44) (44) 129 129 204 32 42 32 + 97 = 28 + 101 = 43 75 53 43 + 86 = 129 39 + 90 = 129
(44) (44) 129 129 248 54 64 54 + 75 = 50 + 79 = 65 119 75 65 + 64 = 129 61 + 68 = 129
(44) (44) 76 86 76 + 53 = 129 72 + 57 = 129 292 163 87 97 87 + 42 = 129 83 + 46 = 129
(44) (44) 129 129 336 98 A8 98 + 31 = 94 + 35 = A9 207 B9 A9 + 20 = 129 A5 + 24 = 129
Table A 4.1. The first two- ` O
160 + 336 = 496 31+ 207 = 248 M 10 31 M 31 = 00; 31 M 20 = 11 204 + 292 = 496 75+ 163 = 248 M 10 75 M 53 = 22; 75 M 42 = 33 248 + 248 = 496 119 + 119 = 248 - 10 119 M 75 = 44; 119 M 64 = 55 163 M 97 = 66; 163 M 86 = 77 207 M 119 = 88; 207 M 108 = 99 496 M 238 = 2 x 129
Table A.4.2. The determinations by the third perfect number, the number 496
234 REFERENCES Dragovich, B., Dragovich, A. p-Adic modeling of the genome and the genetic code, The Computer Journal, 53 (4), 432-441, 2010; (Available also at: arXiv:0707.3043v1 [q-bio.OT]). LR )' ($' ' K (' L )/ ) cheskom kode. Dokl. Akad. Nauk SSSR. 223, 471-474, 1975. Kyte, J., Doollittle R. F. A simple method for displaying the hydropathic character of a protein. J. Mol. Biol. 157, 105-132, 1982. Marcus, S. Symmetry in the simplest case: the real line. Comput. Math. Appl. 17, 103-115, 1989. Négadi, T. The genetic code degeneracy and the amino acids chemical composition are connected, Neuroquantology, Vol. 7, 1, 181-187, 2009; (Available also at: arXiv:0903.4131v1 [q-bio.OT]). U 2*' U duit par Albert Rivaud: Platon, Ouevers completes, tome X, Paris, 1970. Popov, E. M. Strukturnaya organizaciya belkov. Nauka, Moscow (in Russian), 1989. )64' - - # /-tRNA synthetases in correspondence with the codon path cube, Bull. Math. Biol., 59, 645-648, 1997. )64' -- ' H 0 Natural Sciences on Montenegrin Academy of Sciences and arts (CANU), 13, 273-294, 2000; (Available also at: arXiv:q-bio/0611004v1 [q-bio.BM]). )64' - - * ( 4' 97? M 465, 2004a. )64' - - D ; $ TR ` O I D-T Transactions (Faculty of Mechanical Engineering, Belgrade, Serbia), 32, 95-98, 2004b; (Available also at: arXiv: math/0611095v1 [math.GM]) )64' - - $ / ' X=IF>7 >>99 AI-bio.OT], 2006. )64' - - $ = / tereochemical determinism and pure chance, arXiv:0904.1161v1 [q-bio.BM], 2009. )64' -- $ = / / ' X= >: 445 AI- bio.OT], 2011a. )64' -- / ble Knowledge Federation Dialog Belgrade 2011: Partial vs Holistic Oriented Approaches, Sept. 25/ Symposium of Quantum- Informational Medicine QIM 2011, Belgrade, 23-25 September, 2011b. )64' - -' *)4' 447 D / amino acids: synchronic determination with chemical characteristics, atom and nucleon number. J. Theor. Biol. 183, 345 M 349. Shcherbak, V. I. Sixty-four triplets and 20 canonical amino acids of the genetic code: the arithmetical regularities. Part II. J Theor. Biol. 166, 475-477, 1994. Spinadel, V.W. de. The family of metallic means, Visual Mathematics,1, No. 3, 1999. http//members.tripod.com/vismath1/spinadel/. Stakhov, A. P., The Golden section in the measurement theory, Computers Math. Applic. 17, pp. 613- 638, 1989. Swanson, R. A. unifying concept for the amino acid code. Bull. Math. Biol. 46, 187-207, 1984. Sukhodolets, V. V. A sense of the genetic code: reconstruction of the prebiological evolution stage, Genetika, XXI, 10, 1589M1599, 1985 (in Russian). Woese, C.R. et al. On the fundamental nature and evolution of the genetic code. In: Cold Spring Harbor Symp. Quant. Biol., 31, 723-736, 1966.
235 Supplement 2
Golden and Harmonic Mean in the Genetic Code
In previous two works [1], [2] we have shown the determination of genetic code by golden and harmonic mean within standard Genetic Code Table (GCT), i.e. nucleotide triplet table, whereas in this supplement we show the same determination through a specific connection between two tables M of nucleotide doublets Table (DT) and triplets Table (TT), over polarity of amino acids, measured by Cloister energy. [From Proceedings of the 2nd International Conference "Theoretical Approaches to BioInformation Systems" (TABIS.2013), September 17 M 22, 2013, Belgrade, Serbia.]
1. Introduction
In a previous work we have shown that golden mean is a characteristic determinant of the genetic code (GC), regarding on the codons binary tree, 0M63 [1]. In a second one we showed a splitting of Genetic Code Table (GCT) into three equal and significant parts, using the harmonic mean [H(a, b) = 2ab/(a + b); a = 63, b = 31.5]) [2]. In this supplement, however, we will show that a specific unity of golden mean and harmonic mean appears to be the determinant of Rs Table of 16 nucleotide doublets [3] (Tables 1 & 2 in relation to Tables 3 & 4).
01. G GG (6) 02. F UU (4) 03. L 01. G GG (6) 02. F UU (4) 03. L 04. P CC (6) 05. N AA (4) 06. K 04. P CC (6) 05. N AA (4) 06. K 07. R CG (6) 08. I AU (4) 09. M 07. A GC (6) 08. Y UA (4) 09. St. 10. A GC (6) 11. Y UA (4) 12. St. 10. R CG (6) 11. I AU (4) 12. M 13. T AC 14. H CA 15. Q (5) (5) 13. V GU (5) 14. C UG (5) 15. W 16. V GU (5) 17. C UG (5) 18. W 16. T 17. H 18. Q 19. S 20. D 21. E AC (5) CA (5) UC (5) GA (5) 19. L CU (5) 20. S AG (5) 21. R 22. L CU (5) 23. S AG (5) 24. R 22. S UC (5) 23. D GA (5) 24. E Table 1. R Table 2. R
As we have shown, golden mean "falls" between the 38th and 39th codon (38. CAA, 39. CAG), which code for glutamine (Q), a more complex of only two amide amino acids (AAs); two codons, adjacent to the codons (40.UGU, 41.UGC), which code for one of the only two sulfur AAs, cysteine (C). This "harmonization" of diversity is increased by the harmonic mean, in position 42 on the sequence 0-63. The harmonization extends further to "stop" codon (42.UGA) and to codon (43.UGG) that codes for the most complex AA, tryptophan (W). (The "42" as ending position on the "Golden route" M with Fibonacci numbers M on the Farey tree, corresponding with six-bit GC binary tree [1].) On the other side, the splitting of GCT into three parts through harmonic mean [2] makes that AAs are distinguished on the basis of the validity of the evident regularities of key factors, such as polarity, hydrophobicity and enzyme-mediated AAs classification (with parameter values as in Table 2.1 in Rako64' > ?).
236 2. A new rearrangement of nucleotide doublet Table M R 3" 8 # U result: if at the beginning of first sub-system,1 with 6/4 hydrogen bonds, are GG/UU doublets, chemical reasons require GU/UG doublets at the beginning of the second sub-system, with 5/5 hydrogen bonds, instead of AC/CA as it is in Table 1. From the same reasons, we have the changes: CG/GC & UC/CU on the left and AU/UA & GA/AG on the right. With the four first doublets we have four outer squares, i.e. codon families (n1 = GG,UU,GU,UG) in nucleotide triplets Table (TT) which code for nonpolar AAs; the four second doublets give four inner codon 2 families (n2 = CC,AA,AC,CA), which code for polar AAs (Table 3). With the four third (n3 = GC, CU;UA,AG) and four fourth (n4 = CG,UC;AU,GA) doublets are chosen eight intermediate ' A3ES'S'$E8 3.' J' -' 8B A3SE'S'E$' $' $8 30'K' ' "' T8B 3 98 # polar/nonpolar distribution of squares in Table 3 is realized as 4±0 and in Table 4 as 4±1. By this, the polarity/nonpolarity is taken after cloister energy as in Ref. [4].3
2nd letter 1st 3rd U C A G UUU UCU UAU UGU U UUC F UCC UAC Y UGC C C U UUA UCA S UAA UGA CT A UUG L UCG UAG CT UGG W G
CUU CCU CAU CGU U CUC CCC CAC H CGC C C CUA L CCA P CAA CGA R A CUG CCG CAG Q CGG G
AUU ACU AAU AGU U AUC I ACC AAC N AGC S C T A AUA M ACA AAA AGA A AUG ACG AAG K AGG R G GUU GCU GAU GGU U D GUC GCC GAC GGC C V A G G GUA GCA GAA GGA A E GUG GCG GAG GGG G
Table 3. Distributions of AAs after nucleotide doublets presented in Table 2: Four squares with dark tones (outer) contain four first doublets from Table 2 and four light (inner) contain four second doublets.
1 Two subsystems, each with two quadruplets; in total four quadruplets: two on the left as one-meaning (each nucleotide doublet codes for one AA) and two on the right as two-meaning (each nucleotide doublet codes for two AAs or for one AA and termination in the protein synthesis). 2 Histidine (H) is neutral in cloister energy with the value ± 0 [4], but polar in hydropathy [5], polar requirement [6], [7] and in hydrophobicity [8], [9]. 3 E / O / 3^ /8 r of the amino acid from the outside of a G J / / /-philicity because it is an in situ / P A9B
237 In amino acids (within their side chains) at outer/inner areas there are 369/369 nucleons and 61/61 atoms, respectively. All AAs in outer area are nonpolar whereas those in inner area are polar, measured by cloister energy.
2nd letter 3rd 1st U C A G UUU UCU UAU UGU U UUC F UCC UAC Y UGC C C U UUA UCA S UAA UGA CT A UUG L UCG UAG CT UGG W G
CUU CCU CAU CGU U CUC CCC CAC H CGC C C CUA L CCA P CAA CGA R A CUG CCG CAG Q CGG G
AUU ACU AAU AGU U I S AUC ACC AAC N AGC C T A AUA ACA AAA AGA A M R AUG ACG AAG K AGG G GUU GCU GAU GGU U D GUC GCC GAC GGC C V A G G GUA GCA GAA GGA A E GUG GCG GAG GGG G
Table 4. Distributions of AAs after nucleotide doublets presented in Table 2: Four squares with dark tones contain four third doublets from Table 2 and four light contain four fourth doublets; two and two doublets on the right, and two and two on the left. In amino acids (within their side chains), in right/left areas there are 369/369-33 nucleons and 59/58 atoms, respectively. All AAs on the right together with D & E from the left are polar and other on the left are nonpolar. Because the balance is realized in relation to diagonal and not in relation to the type of nucleotide doublets (third or fourth) it follows that positions 3 & 4 in DT are not invariant, but only positions 1 & 2 as it is shown in Table 3.
3. Particles number balances through polarity
Distinctions through polarity, presented in Tables 3 & 4, are followed by the balance of the number of nucleons and atoms. Irrespectively of the Table of nucleotide doublets, V. shCherbak showed [Ref. 10, Fig. 10, p. 173] that it makes sense to display the Table of nucleotide triplets (TT) exactly as here in Tables 3 & 4: four squares at the corners and four squares in the center as ?C ' I ' O#P ' 9 shE )R #' U= ur squares in the corners of TT as well as AAs in four squares in the center of TT have 369 nucleons [(F91 + L57 + V43 + G01 + W130 + C47 = 369); (P41 + T45 + K72 + N58 + Q72 + H81 = 369)].
238 X shE )R insight, we now add: the same quantity give the AAs in right site of TT; the right site in relation to the diagonal FMG4 in TT (S31+Y107+R100+S31+R100 = 369). On the left side of the diagonal there are 336 nucleons (L57+I57+M75+A15+D59+E73=336), what means 33 nucleons less, in relation to 369. With this emergence of difference of "33" on the scene appears a specific self-similarity because the number 33 is an important determinant of the number of atoms in the rows and columns of GCT, i.e. of TT.5 To this self-similarity determination by nucleon number we now also add the self-similarity determination by atom number: AAs in four squares in the corners as well as in the center of TT have 61 atoms in amino acid side chains [(F14+L13+V10+G01+W18+C05=61); (P08+T08+K15+N08+Q11+H11=61)]. In relation to the diagonal FMG, in TT, there are 58 and 59 atoms, respectively; on the left: L13+I13+M11+A04+D07+E10=58, and on the right: S05+Y15+R17+S05+R17=59. These quantities (58 and 59) are the same as the quantities of / 0) R / (what is a further self-similarity): 58 in two inner and 59 hydrogen atoms in two outer rows [Ref. 11, Tab. 7, p. 830], [12]. (Notice that 58 + 59 = 117 is total number of hydrogen atoms in 20 canonical AAs of GC, within their side chains, what is the self-similarity once more.)
4. Determination through "golden whole"
The splitting of GCT (i.e. TT) into 4 outer, 4 inner and 8 intermediate squares, corresponding to responsible nucleotide doublets, leads us to the following conclusion. Within the set of all n- gons, where n is even number, the case n = 4 is only and one case where harmonic mean of "golden whole" (n2 M n)6 and its half [(n2Mn)/2] equals 2n, and n2Mn = 3n. So, in this case we have that the ratio 2:3 appears to be the harmonic mean within the harmonic mean, and, by this, the sequence n1 M (n3 or n4) M n2 corresponds with the Cantorian triadic set. Moreover, such a harmonic mean appears to be corresponding with the number of "small squares" within intermediate space in form of only one "ring" as it follows: [(2 + 2) + (4 x 0) = 4]; [(4 + 4) + (8 x 1) = 16]; [(6+6) + (12 x 2) = 36]; [(8+8) + (16 x 3) = 64] etc. As it is self-evident, the symmetrical "out M middleM in" arrangement (1:1:1 of rings) is not possible for n + 9' n-gons nor for n-letter alphabets. At the same time here is a self-similarity expressed through the
4 Starting from diagonal (FMG) together with two adjacent ones (SMR and LMT8 # 3OP 8 in a strict balance with the set of polar AAs, polar through hydropathy [5B= # # 3OP' OP F ' 8 U/ / / >' respectively [Ref. 11, Tab. A.3, p. 840 in relation to equations 11-14, p. 838]. 5 If we consider the set of "61" of AAs, then in two rows, YNR & RNY, there are 8 x 33 and in two other, YNY & RNR, 10 x 33 atoms. On the other hand, in two pyrimidine columns, NYN, there are (9 x 33) M 1 and in two purine ones, NRN, (9 x 33) + 1 of atoms [Ref.12, Tab. 3a, p. 224]. (Y for pyrimidine, R for purine and N for all four types of nucleotides.) 6 I O # P U = n2± n ^ OM[G.TPC n = 4, the difference is 12 and the sum 20. In relation to genetic code there are self-similarities expressed through mathematical operations: (4 x 4) M 4 = 12; (4 x 4) + 4 = 20, (4 x 4) x 4 = 64. [Number 12, correspondent to 4+4+4 doublets in sequence [n1 M (n3 or n4) M n2]; number 20 as 20 AAs (4 x 4 = 16 AAs of alanine stereochemical types and 4 of non-alanine stereochemical type); number 64 as 64 codons.] (About four stereochemical types see Ref. [13] & [14].)
239 number of "small squares" in the sequence n1 M (n3 or n4) M n2 and the number of codons within them: four squares per n1, n2, n3, n4, each square per four codons. Moreover, there is a self- similarity between golden and harmonic mean versus 4-letter alphabet: 1n as 1 square (1 nucleotide doublet), 2n as harmonic mean (in the sense above said), 3n as golden whole and 4n as the sum n1+ n2+ n3+ n4; all these versus 1 letter of alphabet (as letter minimum), 2 letters as word root (nucleotide doublet), 3 letters as 3-letter word (codon) and 4 letters as letter maximum within alphabet.7
5. Concluding remark
With the title of the supplement is given a working hypothesis that the golden mean (GM) and harmonic mean (HM) are determinants of the genetic code. The findings presented by four illustrations show that this hypothesis is confirmed. However, unlike the previous access to the same determination, it refers not only the analysis of the nucleotide triplet Table, but rather refers to the two tables M Table of doublets (DT) and Table of triplets (TT). In fact, it is precisely presented that these tables are unique in terms of determination just over GM and HM. It is expected that all these uniqueness correspond to the same, or similar uniqueness, found by other authors ([16], [17], [18], [19]), what in future researches should be checked. However, presented facts are such that ones reaffirm the other and vice versa. All together, they favor the recognition that the chemical reactions that determine the GC are not only the reactions in a "test tube", but these reactions are associated with a specific balance of the number of particles (atoms and nucleons); balance, determinated by unique arithmetic and algebraic regularities and expressed in the form of specific (nonfractal) self-similarity ("a harmonized chemistry"). From this it follows further that presented facts also support the hypothesis that the genetic code was from very begining, in prebiotic times and conditions, a complete code [10], [15]. On the other hand, the knowledge that "the chemistry of living" is actually a harmonized chemistry requires great care in medicine, agriculture and natural environment, taking into account the fact that this harmonization is strictly immanent to the living as such, mediated by genetic code as such.
7 By this one must notice that all these self-similarities are possible only for 4-letter alphabet and 3-letter words.
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