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HN VIỆT DỊCH S LƯỢC

GS Nguyễn Hữu Quang

Nguyn Giảng Vin Vật L Chuyn về Cơ Học Định Đề
(Axiomatic Mechanics, a branch of Theoretical Physics)
tại Đại Học Khoa Học Si Gn trước năm 1975

 

 

 

CHƯƠNG 05

 

DỊCH TIỀN ĐỀ

 

 

(Tiếp theo Kỳ 49)

 

FURTHER READING

           Two
            
_______________________________________________

CONGRUENCE FUHSI-WATSON-CRICK 

                                  易。

 

 Change is recursive and cross-genetic.

                                              − The Great Appendix, Section I, V/6

The physicist Erwin Schrdingers book What is life? (1944) had a great impact on the development of molecular biology and stimulated scientists such as Watson and Crick (E).

The problem of relating digrams to the codons and the genetic code to the hexagrams was raised by the molecular biologist Gunther Siegmund Stent (1924-2008) in his book The Coming of the Golden Age; a view of the end of progress (2G) and, four years later, by Dr. Martin Schonberger in his book The I Ching and the Genetic code; the Hidden Key to Life (2F). The former mistook the minor yin for the minor yang because of a misinterpretation by Richard Wilhelm himself in his translation of the original I CHING or Book of Changes (2I && 2J) into the German I Ging. The latter mixed up everything and his congruence between digrams and triplets fails the horizontal test of a function. Finally, came Dr. J. F. Yang with his DNA and the I Ching, the Tao of Life (2K). His interpretation conforms to the tradition of the Four Emblems and agrees with my own. 

It is important to notice at once that di-, tri- and hexagrams grow upwards like trees when alone and outwards like trees yearly rings when grouped in cyclads. Hence, when writing down or reading out a hexagrams binary representation, we do it from left

Consequently, the readers may notice that the four emblems are: Old Yang :, Lesser Yin ;, Lesser Yang <, Old Yin =, and not the other way around. Thus, in the Lesser Yin, the yin line is younger than the yang line because it is in a higher (later) position than the yang line. Similarly, in the lesser yang <, the yang line is younger than the yin line for it is in a higher position than the latter. Thus, we have the following table:

Table 2.1 Digrams and their Properties

Digram

Bin.

Val.

Dec.

Parity

Mantic

Value

Emblem

Nucleotide

Base

Sym

=

00

0

even

6

Old or

Major Yin

Adenine

A

< 

01

1

odd

7

Young or

Lesser Yang

Cytosine

C

;

10

2

even

8

Young or

Lesser Yin

Thymine

(Uracil)

T

U

:

11

3

odd

9

Old or

Major Yang

Guanine

G

                 DNA (deoxyribonucleic acid) is the fundamental substance of which genes are composed. Chemically, it is a double chain of linked complementary nucleotides (A-T or C-G), having deoxyribose as their sugars.

            On the other hand, RNA (ribonucleic acid) is a single-stranded nucleic acid similar to DNA but having ribose sugar rather than deoxyribose sugar and uracyl rather than thymine as one of the bases. The principal function of RNA is to copy the genetic information from the DNA and then to transform into proteins. Chemically, RNA differs from DNA by an additional hydroxyle -OH linked to the pentose.

The five nucleotide bases have the following chemical structures, formulas and molecular weights (mw):

 

     Thymine          Uracil            Adenine      Cytosine         Guanine

    (2,6 dihydroxy-  (2,6 dihydroxy-  (6-aminopurine) (2-hydroxy-     (2-amino-

       5-mehtyl-           pyrimidine)                      -aminopyrimidine)  6-hydroxypurine)

       pyrimidine)  

    C5H6N2O2         C4H4N2O2          C5H5N5         C4H5N3O          C5H5N5O

     mw = 126.12    mw = 111.9     mw = 135.13    mw = 111.10      mw = 151.13

                  Fig. 2. 2 Structure and Molecular Masses of the Nucleotide Bases 

         Cytosine and thymine (uracil) are pyrimidines with two hydrogen-bonds thanks to the non-paired electrons of the nitrogen atoms; adenine and guanine are purines with three H-bonds as shown in the following chemical structures:

        Fig. 2.3 Pyrimidines Structure                         Fig. 2.4 Purines Structure

 The assignement of the 4 nucleotide bases to the 4 digrams is natural and consistent for several reasons:  

            1) In Yilogy, the odd numbers are yang and the even numbers are yin so that the yin/yang property of the digrams matches that of the enblems. This property corresponds to the Watson-Crick pairing of A to U (A-U) and of G to C (G-C).

2) The purines (A, G) are classified 'major' and the pyrimidines (C, T/U) 'minor' because the formers are larger in dimensions and heavier in weights than the latters: G = 151.13 > C = 111.10; A = 135.13 > T = 126.12 and U = 111.9.

            3)  The Watson-Crick pair A-U(T) is yin or even, with an even number (2) of H-bonds while the pair G-C pair is yang or odd with an odd number (3) of H-bonds. Moreover, G-C pairs are denser than A-U pairs and the G-C content is usually estimated by measuring the density of DNA double helices.

            4) Leading zeroes are meaningless; the leading poly-A (0) (polyadenylic acid) do not encode any amino acid; but once initiated, the codon AAA = B (Kun) encodes lysine. By the way, in the amino acid composition of proteins, lysine appears very frequently. This suggests that there is excess of 0s not replaced by 1s. After the initiation by AUG = u (progress) = methionine (Met), the leftover poly-A contribute to the excess of lysine.

5) Chemically, the codon GGG = A (Chien) and its anticodon CCC = (Wei Chi) (i.e. the first and the last hexagrams in the King Wns sequencing) form a total of 9 H-bonds. On the other hand, the codon AAA = B (Kun) and its anticodon UUU = ? (TTT) (Chi Chi) (the second and the antepenultimate hexagrams in the King Wns sequencing) form 6 H-bonds between them. The remained codon-anticodon pairs could possibly have 2 + 2 + 2 = 6, 3 + 2 + 2 = 7, 3 + 3 + 2 = 8 or 3 + 3 + 3 = 9 H-bonds, i.e., one of the very four mantic numbers of the digrams 6, 7, 8, 9. As a matter of fact, in the 3-coin method of divination, Head (yang) has a value of 3 and tail (yin) a value of 2, following tightly the syntagma: The number 3 was assigned to Heaven, the number 2 to earth, and from them came all the other numbers. (Discussion of the Trigrams, I/2). 

Remember that a protein is a macromolecule of amino acides chained end to end with a linear string. The general formula for an amino acid is H2N―CHR―COOH, in which the lateral chain, or R group, can be anything from a single H atom (as in glycine Gly) to a complex ring (as in tryptophan Trp):

Here are the side-chains of some of the remained amino acids along with their 3-letter abbreviations: 

           There are 20 common amino acids in living organisms (Table 2.1.4), each having a different R group. Amino acids are linked together in proteins by covalent bonds called peptide bonds which are formed through a condensation reaction involving the removal of an H2O molecule:

         n other words, the -NH2 group of one amino acid interacts with the -COOH of a

 

A peptide possessed a zigzagging backbone of nitrogene and carbon analogous to the sugar-phosphate backbone of a polynucleotide with R groups projecting outward in alternating fashion. A peptide containing many peptides, is called polypeptide. 

          Table 2.2 The 20 common Amino Acids in living organisms

The amino acids with asterisks are hydrophobic; the rest are hydrophilic. To this list we should add the Stop codon (symbol X) or termination codons. They are regarded as similar to periods or commas punctuating the message encoded in the DNA.

            There are mainly four classes of RNA: tRNA = Transfer RNA (it carries the information contained in a gene to the ribosome); rRNA = Ribosomal RNA (component of the ribosomes that serves as a kind of scaffolding for the processes of polypeptide synthesis); mRNA = Messenger RNA (the mRNA molecules, which are produced in the DNA template, pass out  through the nuclear pores into the cytoplasm and here the information in the sequence of the mRNA is translated into protein), and hetRNA = heterogenous nuclear RNA (found only in eukaryotes).

          If a single nucleotide in an mRNA specifies a single amino acid in a protein, it would mean that proteins could contain only four amino acids, whilst in living organisms they contain 20. Similarly, a doublet code could only generate 42 = 16 combinations of pairs of nucleotide bases, still not enough to specify all 20 amino acids. Therefore, we need 64 triplet codes. 16 < 20 < 64 shows that the code is degenerate, meaning that an amino acid may be specified by more than one triplet. In fact, as many as 64 20 = 44 of the putative triplets are never found in natural mRNAs.

Thus we have exactly 64 triplets because each time we can chose one out of the four letters G, U, C, A; since there are three choices for the codons, we get, all in all 4 x 4 x 4 = 64, the very cardinality of the set of hexagrams. The triplet codes of a codon or anticodon correspond to the three digrams of a hexagram. When poly AC is synthetized from mixture containing equal proportions of A and C, eight triplets should occur: AAA, AAC, ACA, CAA, ACC, CAC, CCA, and CCC. Using poly AC it was found that six amino acids are incorporated into polypeptides: asparagine (AAC), glutamine (CAA), histidine (CAC), lysine (AAA), threonine (ACC & ACA), and proline (CCC & CCA).  

Now its about time to establish the congruence between the Watson-Crick Genetic code and the Fushian biocode through the following Tables 2.2 and 2.3, in which we adopt the following conventions: 

         In Table 2.3 The Genetic Code, the letters are proceeded from left to right as usual;

         In Table 2.4 The Fuhsian Biocode, we respect the tradition by sequencing the digrams from bottom to top: the first digram (Earth-digram) is at the bottom, the second digram (Man-digram) is in the middle and the third digram (Heaven-digram) ends up at the top.

By the way, it is good to notice the three digrams positions in a hexagram reflect the nature of the genetic DNA-RNA-protein trio: DNA as Earth (Space), RNA as Heaven (Time) and protein as Man (Humanity). This kind of triad abonds: 

Biology(DNA)    Western          Chinese           Yiching           Hinduism         Modern

                       Philosophy       Philosophy                                                  Science

Codon              Mind               Heaven           Pre-Heaven       Brahma          Information

Pscychon          Soul                Mankind         In-Heaven         Vishnu            Energy

Somaton          Body                Earth             Post-Heaven      Siva               Matter

Exercise: Find the triads corresponding to: Buddhism, Christianism, Neo-Confucianism, Computer Science, and Psychology.                                             

Any hexagram comprises either 3 digrams (top, median, bottom) or 2 trigrams (upper and lower). Traditionally, it is read top down. Its why, in the Fufsian biocode the order is: top, median, bottom                       

Table 2.3 The Watson-Crick                                       Table 2.4 The Fuhsian

                      Genetic Code                                                      Holistic Biocode  

The table 2.3 reveals several features of the genetic code: 

         The code is seen to be degenerate: six codons are used to specify a single amino acid (arginine, leucine or serine);

         Only two amino acids are represented by a single codon: tryptophane UGG and Methionine AUG. The latter specifies the initiation of translation;

         The three codons UAG (amber codon), UGA (opal codon), and UAA (ochre codon) are often called nonsense because they fail to stimulate acyl-tRNA binding and serve only as chain-terminators signals. 

Two notable patterns emerge from this table 2.3: 

1.     Certain amino acids can be brought to the ribosome by several tRNA species having different anticodons while certain other amino acids are brought to the ribosome by just one tRNA. This feature of the code is thought to minimize the consequences of errors made during translation or of mutagenic base substitutions.

2.     Certain tRNA species can bring their specific amino acids in response to several codons, not just one, through a sloppy pairing at one end of the codon and anticodon (wobble). For instance, all codons starting with AC specify threonine, all codons starting with CC specify proline, all codons starting with CG specify arginine, all codons starting with CU specify leucine, all codons starting with GC specify alanine, all codons starting with GG specify glycine, all codons starting with GU specify valine and all codons starting with UC specify serine. This flexibility in the nucleotides of a triplet may well help to minimize the consequences of errors. At the same time, the upper trigram of the corresponding hexagrams remains the same (specifically !chien, -kn, )sun, %li, #tutui, or 'chn). Here the names of the trigrams are not capitalized to differentiate them from the hexagrams having the same names. 

I have asked the mathematician Brian Hayes how to calculate the total number of possible genetic codes. Here is his answer:

There are 20 amino acids plus the "stop" signal, so each of the 64 codons can have any of 21 possible meanings. Thus a first approximation says there are 2164 possible genetic codes; this number is equal to 4.1882719x1084. However, not all of those codes are legal. We require that every amino acid and the stop signal have at least one codon assigned to it, and so we have to exclude all of the codes that do not satisfy this condition. How many of those illegal codes are there? Well, if we choose some specific amino acid, say ALA, there are 2064 codes that do not assign any codon to ALA. It's the same for ASN, ASP, etc. Thus there are 21 x 2064 codes that fail to include at least one amino acid or the stop signal. Therefore the number of legitimate codes is 2164 - (21x2064), which works out to 3.144556x1083. Here is the exact number, in case it might amuse you: 

314455595548268427310020743752976432915773724299403513508038881183728210190544679681.

I figured out that this result is obtained by using the software Derive 6.1. On the other hand, the exact number of possible genetic codes is:

4188271851027274266670020743752976432915773724299403513508038881183728210190544679681
@
4.188271851 x 1084. 

 

Table 2.5 King Wns Arrangement

 

N. B. In the Table 2.5, the octal representation of the King Wns hexagram precedes the period and the binary value of the Fushis hexagram follows it. The name of the corresponding hexagram is given in Enhanced Wade-Giles. Last last row gives the tricode as well as the single-letter of the corresponding amino-acid given in Table 2.2.

HEXAGRAMS DERIVATION 

An r-permutation anr (r b n) of n things is an ordered arrangement of r of them. In any book on Combinatorial Mathematics, it is proved that there are

(n)r =  n(n 1) (n -2) (n r + 1)

r-permutations of n-objects (0 < r b n). The symbol (n)n , written n!, read "n-factorial ", and (n)r is sometimes called the falling r-factorial of n. 

            It is also be proven that if there are

n = n1 + n2 + + nk

objects, of which n1 are of one kind,  are of a second kind, , nk  are of a kth kind, then the number of permutations of the n objects is

          Finally, it is proved that the number of r-subset of a set S containing n elements is

            When there are only two labels we can shorten the notation and write


 

Pascal's triangle is a number triangle with numbers arranged in staggered rows such that

where  is a binomial coefficient. The triangle was studied by the French mathematician, physicist, philosopher and writer Blaise Pascal (1623-62), although it had been described centuries earlier by the Persian astronomer-poet Omar Khayym (1048-1131), who helped reform the ancient Muslim calendar, and by the Chinese mathematician Yang Hui Dương Huy (c. 123998), active in the great flowering of Chinese mathematics during the Southern Song Dynasty (1127-1279) Thus, it is known as the Yang Huis triangle in China.

            In his book, A New Kind of Science (p. 611), Stephen Wolfram has presented the Pascals triangle as nested patterns constructed using arithmetic operation. He also generalized this triangle so that each number is the sum of the numbers above it and to its left and right on the row above.  

The Pascals triangle starts with n = 0:

             Fig. 2.5 Yang Huis Isoceles Triangle           Fig. 2.6                  Fig 2.7

                                                                                    Pascals Rectangle Triangles

N.B. In Fig. 2.7, the digits A, F and K are successively the decimals 1010, 1510, and 2010 in a number system of radix 21 (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, G, H, I, J, K). In that case, the different lines of Fig. 2.4.2 are simply the successive powers 0, 1, 2, 3, 4, 5, 6 of the number 1110.

Pascal's formula shows that each subsequent row is obtained by adding the two entries diagonally above,

For n = 2, the binomial becomes:

                                    (a + b)2 = a2 + 2ab + b2

Replacing a and b by the monograms ? and > meaning 0 and 1, we get

the four forms:

 

a2            00  old yin                       moving yin                    Adenine (A) 

2ab       =  01  lesser yang                resting yang                  Cytosine (C)

            =  ;  10  lesser yin                   resting yin                     Thymine (T)

b2         =  11 old yang                     moving yang                  Guanine (G)

          For n= 3, the binomial becomes:

(a + b)3 = a3 + 3a2b + 3 ab2 + b3

Replacing a and b by the monograms ? and > meaning 0 and 1, we

get the eight trigrams:

 

a3         =  / 000  08   -0         kun      Ultimate Yin                                         

3a2b      =  - 001  18   -1         kn       Lesser Yang    

             =  + 010  28   -2         kan      Younger Yang

             =  ) 011  38    -3        sun       Older Yin       

3ab2      =  ' 100  48    +0        chn     Older yang     

               =    % 101   58  +1        li           Younger Yin   

             =  # 110  68  +2        tui         Lesser Yin       

b3         =  ! 111  78   +3        chien    Ultimate yang  

N.B. The first number is binary, the second one octal, and the third one algebraic. The trigrams names are not capitalized in order to distinguish them from the hexagrams having the same names. In the last column we give the aliases of the corresponding trigrams in the terminology of the Pristine Envelop Classic. Here the leading letters are capitalized in order to distinguish the trigrams from the digrams having the same nanes.  

For n = 6, the binomial becomes: 

           (a + b)6 = a6 + 6a5b+ 15a4b2+ 20a3b3 + 15a2b4 + 6ab5 + b6

In the last column, the mantic value is calculated by the general formula:

(2n - 1) x (5 1) with n = 1..6 (line number)

The "+" sign corresponds to a yang line > and the "-" sign to a yin line ?.

Replacing again a and b by the monograms ? and >, we get:

 6G    bin     octal   trinucleotide               mantic value =

a6        = B 000000 00            AAA                         144 

6a5b    = X 100000 40             AAU                         146

            G 010000 20             AAC                         150

            O 001000 10             AUA                         154

            P 000100 04             ACA                                 158

            H 000010 02             UAA                         162

              W 000001 01              CAA                        166

15a4b= S 110000 60             AAG                         152

            d 101000 50              AUU                         156

            s1100100 44            ACU                         160

            C 100010 42              UAU                         164

            [ 100001 41              CAU                         168

            n   011000 30           AUC                         160

            h 010100 24              ACC                         164

            ]1010010 22             UAC                         168     

            D1010001 21             CAC                         172

            ~ 001100 14              AGA                         168

            g 001010 12              UUA                         172

            t 001001 11              CUA                         176

            m 000110 06              UCA                         176

            c 000101 05              CCA                         180

            T 000011 03              GAA                         184

20a3b= K 111000 70             AUG                         162

            v 110100 64              ACG                         166

            i 110001 61              CAG                        174

           w1101100 54             AGU                        170

            101010 52              UUU                        174

            V 101001 51              CUU                        178

            Q 100110 46              UCU                        178

            U 100101 45              CCU                        182

            j 100011 43              GAU                        188

            ` 011100 34              AGC                       174

            p 011010 32              UUC                        178

            R  011001 31             CUC                        182

            010101 25              CCC                        188

            { 010011 23              GAC                        190

            _ 001110 16              UGA                        186

            x 001101 15              CGA                        190

            u 001011 13              GUA  `                            194

            L 000111 07              GCA                        198

15a2b4 b 111100 74             AGG                        176

            E 111010 72              UUG                       184

            Z 111001 71              CUG                        184

            z 110110 66              UCG                        188

            f 110101 65              CCG                        178

            } 110011 63              GAG                        192

            q 101110 56              UGU                       188

            ^ 101101 55              CGU                        192

            e 101011 53              GUU                       196

            Y 100111 47              GCU                        200

            r 011101 35              CGC                        196

            y 011011 33              GUC                        200

            F 010111 27              GCC                        204

            a 001111 17              GGA                        208

6ab5  =   k 111110 76              UGG                       194

           N  111101 75              CGG                       198

            I 111011 73             GUG                       202

            J 110111 67              GCG                       206

            M 101111 57              GGU                       210

            L 011111 37              GCA                        198

b6        = A 111111 77              GGG                       216

Mark White, a physician and inventor in Bloomington, Indiana, discovered the genetic code can be represented succinctly on an icosidodecahedron which is a combination of a dodecahedron (made up of 12 pentagones) and its dual the icosahedron (made up of 20 triangles). Each pentagonal face of this solid is labeled with one of the four nucleotides, each of which appears thrice. Each encoded amino acid will be found adjacent to the first base, along the edge leading to the second base, and inside the triangle formed with the third base. 

         Fig. 2.8 Icosidodecahedral Representation of the genetic code discovered by Mark White and embodied by him in a G-Ball (G stands for Genetic code), a.k.a. Rafiki Map.

  For example, in Fig. 2.4, a tour from A to A to U yields asparagin, a tour from  A to G to U  gives serine, and a tour from A to U to G specifies methionine. 

In the following figures, we can identify successively lysine (AAA), proline (CCC & CCA), glycine (GGG) and Phenylalanine (UUU).

              Fig. 2.9 G-Ball 1: Lysine (AAA)              Fig. 2.10 G-Ball 2: Proline (CCC & CCA)

         Fig. 2.11  G-Ball 3: Glycine (GGG)     Fig. 2.12 G-Ball 4: Phenylalanine (UUU)

White notices that the icosidodecahedral model is closely related to the first coding scheme of the genetic code, the diamond code, proposed in 1954, by the physicist George Gamow, the chief proponent of the Big Bang theory in Cosmology (2D).  This takes us back to square one (Fig. 2.9).   

 

                  Fig. 2.13 The Diamond Code    Fig. 2.14 The 20 classes of the 64 Codons
 

B O     P ~           02 15    16 62

D R   r            04 18    64 50

} I   J A            61 09    10 01

C ?   Q q            03 63    17 49

G W n t           07 23 46 52      

S T K u           19 20 11 53

X H d g           24 08 36 39

h c ` x           40 35 32 56

v L b a           54 12 34 33      

s m w _           51 45 55 31

i { Z y           41 59 26 57

[ ] V p           27 29 22 48      

f F N l           38 06 14 44

U o ^ \           21 47 30 28

j | e E           42 60 37 05

Y z M k            25 58 13 43

Fig. 2.15 The 20 classes       Fig. 2.16 The King Wn
     of 64 hexagrams                  Perspective

We will see in CHAPTER SIX: THE YI ABSTRACT ALGEBRA that the set H of hexagrams is a commutative ring under the modulo operations. Moreover, the congruence between the codons and the hexagrams confers to the formers a similar algebraic structure. Thus, in Fig 2.10, the symmetries of the diamond code sort out the 64 tricodons into 20 classes. All the codons in each class specify the same amino acid. The 16 palindromic tricodons at the top of the table 2.10 come in pairs and correspond to the first 16 hexagrams in Fig 2.15; the remaining tricodons are in groups of four and correspond to the 48 remaining hexagrams.

Table 2.6 Relationship between Quadrilines and Hexagrams 

N.B.        1) In this table, 1 Chi = 7 yuan = 7 x 3 or 21 chi = 7 x 3 x 20 or 420 pu = 420 x 4 or 1680 metonic cycles = 31920 sui;

NOTES

1) The first row contains the the starting Gregorian date (GD) of the leading tsan of the leading shou in each column.

2) The second row, the bottom row and rightmost column give the ternary (in bold) and balanced ternary values (in italics) of the shou situated at their intersection.

3) Notice that the rightmost column contains top down: the row number starting with 0, the balanced ternary values (in bold) and the ternary (in italics).

4) Each cell contains successively the details on each of the 81 shou:

a) its enhanced Wade-Giles romanization (in particular, in the first and the last row of cells, the EWG romanization is preceded by a column number) ; b) Its Chinese name; c) Its ordinal number along with its native representation; d) The ordinal number along with the representation of the corresponding hexagram. 

The interconversion between the ordinal number n of a shou and its balanced ternary value is proceeded as follows:

Since, n є [1, 81]|1 ≤ n ≤ 81. Dividing n by 9, we obtain a quotient q and a remainder r: n = 9 x q + r. Its decimal nonal representation is qr with both q and r in the interval [0, 8]. Now we can uses the phalanges of the index I, the medium M and the ring finger R on both hands as cells of two similar tables for this matter. The two thumbs are pointers. 

r

 

I

M

R

yx

 

> 

?

---

 

 

1111

22

3

> 

1

0

3

6

?

2

1

4

7

---

3

2

5

8

q

 

I

M

R

yx

 

1>

?

---

 

 

1111

22

3

> 

1

0

3

6

?

2

1

4

7

---

3

2

5

 

 

 

 

  

  Left Hand Nonal left digit q         Right hand Nonal right digit

Examples:     N44 n = 43 = 9 x 4 + 7 = 479 = 22|32 =    : ;  =  ;:;

      N67 n = 66 = 9 x 7 + 3 = 739 = 32|21  =     <+ 3  = 3<.

 In examining the systematics of indices of physico-chemical properties of codons and amino acids across the genetic code, Bashford and Jarvis (2A)) used a simple numerical labelling scheme for nucleic acid bases, A = (-1, 0), C = (0, -1), G = (0, 1), U = (1, 0), to fit data as low order polynomials of the 6 coordinates in the 64-dimensional codon weight space. Their work shows that fundamental patterns in the data such as codon periodicities, and related harmonics and reflection symmetries are associated with the structure of the set of basis monomials chosen for fitting.

We will see in Chapter Seven: Triadic Combinatorics and Abstract Algebra, that a quadriline or Shou is formed by the superposition of four triadic lines, i.e. each line can take one of the 3 values --- , --,  and > standing for  or -1, 0 and 1 in the balanced ternary number system (Table 2.6). Since each quadriline has 9 tsan, it may be either embedded in a sextiline or simply included in the quadriline as superscript. Thus, codons are labeled either as ordered sextuplets, or by a superscripted quadriline or sextiline. For example,
 
 
   CGA = (0, -1, 0, 1, -1, 0) :=
3@ <... 

 

 BIBLIOGRAPHY:

2A Bashford, J. D. & Jarvis, P.D. The Genetic Code as a Periodic Table: Algebraic Aspects, submitted to Biosystems, physics/0001066, (http://www.physics.adelaide.edu.au/~jbashfor/).

2B . Systematics of the genetic code and anticode: history, supersymmetry,

        Degeneracy  and periodicity, physics/9809030, in Proceedings of the XII

        International Colloquium on Group Theoretical Methods in Physics, S.P. Corney, R.

        Delbourgo and P.D. Jarvis (Eds.) (International Press, Boston), 147-151.

2C Hayes, Brian. Ode to the Code, American Scientist, Nov.-Dec. 2004, vol. 92, number 6, p. 494.

2D Gamow, G. "Possible relation between deoxyribonucleic acid and protein structure", Nature, vol. 173, p.318, 1954.

2E Murphy, M. P. & ONeill, L. A.J., eds. What is Life? The Next Fifty Years, Speculations on the future of biology, Cambridge University Press, Cambridge, 1997.

2F Schonberger, M. Verborgener Schlssel zum Leben (The I Ching and the Genetic code; the Hidden Key to Life), ASI Publishers Inc., New York, 1979.

2G Stent, G.S., The coming of the Golden Age, a view of the end of progress, the National History Press, Garden City, N.Y., 1969.

2H White, Mark. The G-Ball, a New Icon for Codon Symmetry and the Genetic Code,

         http://www.codefun.com .

2I Wilhelm, Richard. I Ging, 14. Auflage, Eugen Diederichs Verlag, Mnchen, 1990.

2J . The I Ching or Book of Changes, rendered into English by Cary F. Baynes,  Third Edition, Bollingen Series XIX, Princeton University Press, 1967.

2K Yan Johnson F. DNA and the I Ching: The Tao of Life, North Atlantic Books, Berkeley, 1991. 

 

 

   

 

    Xem Kỳ 51

 

 

 

 

GS Nguyễn Hu Quang
Nguyn Giảng Vin Vật L Chuyn về Cơ Học Định Đề
(Axiomatic Mechanics, a branch of Theoretical Physics)
tại Đại Học Khoa Học Si Gn trước năm 1975

 

  

 

 

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