, where we have X4 CYTOSINE :NT{C}=NT{X } = [ ].{4;XXXX} = [10 ].4;TGCA

15.5 DETERMINTION OF NT OF DN MINO ID In page 70, if we include in the utilized concept of Boolean rithmetical Field (BFi) of the previous item, the nucleotide THYMINE ( T ), instead of URIL { U ) as the Boolean rithmetical Variable (BV), that is, X = T instead the utilized X = U for RN of the TBLE 7, we obtain these numerical expressions, but now relative to the DN. Then, we can apply these Boolean rithmetical oncepts into the symbols or letters used in the enetic ode. But now, we have the bases called nucleotides, {T,,, } respectively, Thymine, uanine, ytosine and denine, are considered as new Boolean rithmetical Variables (VBs) in a Boolean rithmetical Field (BFi) whose cardinality number is K= and ordinality number is ω = { X, X, X, X1}, where we have X = T, X =, X = and X1 =.. Then, the Numerical Transforms (NTs) to the these ordered numerical variables, that is in the enetic ode, the Numerical Transforms (NTs) of the nucleotides bases, {T,,, } are the following: 1 1 1 ( ) { } DENINE : NT{}=NT{X } = [1010 1010 1010 1010].{;XXXX} = [10 ].;T ( ) { } YTOSINE :NT{}=NT{X } = [1100 1100 1100 1100].{;XXXX} = [10 ].;T = ( ) { } UNINE : NT{}=NT{X } [1111 0000 1111 0000].{;XXX X} = [1 0 ].;T 1 = 1 = { } THYMINE : NT{T}=NT{X } [1111 1111 0000 0000].{;XXXX} [1 0 ].;T or, NOTE: The Numerical Transforms of these four organic bases of the DN can be presented together as a the following system simultaneous: 1111 1111 0000 0000 1111 0000 1111 0000 NT = = { ;T} 1100 1100 1100 1100 1010 1010 1010 1010 or, in vertical hexadecimal presentation of these four organic bases of DN: () NT = [ FED B98 765 10 ].;T { } It is possible to obtain afterwards all the numerical expressions of a System of three Boolean rithmetical Functions (BFs), which corresponds to each one of 89

the 6 possible codons (or triplets of nucleotides). For example, the codon T, which corresponds exceptionally, to the amino acid called TRIPTOPHN, has the following Numerical Transform, as a system of three BFs: E T Trp (Y 8 ) (TRYPTOPHN) T X X X X 1 T t 0 0 0 0 0 0 0 0 0 t 1 0 0 0 1 0 0 0 0 t 0 0 1 0 0 0 0 0 t 0 0 1 1 0 0 0 0 t 0 1 0 0 0 1 1 t 5 0 1 0 1 0 1 1 t 6 0 1 1 0 0 1 1 t 7 0 1 1 1 0 1 1 t 8 1 0 0 0 1 0 0 t 9 1 0 0 1 1 0 0 t 10 1 0 1 0 1 0 0 t 11 1 0 1 1 1 0 0 t 1 1 1 0 0 1 1 1 7 t 1 1 1 0 1 1 1 1 7 t 1 1 1 1 0 1 1 1 7 t 15 1 1 1 1 1 1 1 7 TBLE 8 Then, we have the following numerical representation to the amino acid TRYPTOPHN of DN: 1111 1111 0000 0000 NT { Trp} = 1111 0000 1111 0000 =.{ ;T }, 1111 0000 1111 0000' or, in vertical presentation: (7) NT { Trp} = = 7 0.;T { } To obtain the Numerical Transforms of the correspondent twenty different amino acids of the DN, which have six four three one codon (that is, the triplets of nucleotides), including the three triplets that do not stands as amino acid, called punctuation (or, terminator), it is convenient to prepare the TBLE 10: In this TBLE the Boolean rithmetic Functions (BFs) of TBLE 6, 90

{ Y 1, Y, Y,..., Y 0}, are replaced by the standard one-letter symbols [18] whose key for DN amino acid sequence is given by the following TBLE 9: MINO ID FOR DN lanine (la) rginine (rg) sparagine (sn) spartic acid (sp) ysteine (ys) lutamine (ln) lutamine acid (lu) lycine ly) Histidine (His) Isoleucine (Isso) Leucine (Leu) Lysine (Lys) Methionine (Met) Phenylalanine (Phe) Proline (Pro) Serine (Ser) Threonine (Thr) Tryptophan (Trp) Tyrosine (Tyr) Valine (Val) ONE LETTER SYMBOLS R N D Q E H I L K M F P S T W Y V TBLE 9 91

E T F S Y L P E W H R I T Q N K M V D Y 1 Y T Phe Ser Tyr ys Leu Pro lu Trp His rg Ileu Thr ln sn Lys Met Val la sp ly (*) (**) X X X X 1 t 0 0 0 0 0 t 1 0 0 0 1 t 0 0 1 0 t 0 0 1 1 t 0 1 0 0 t 5 0 1 0 1 t 6 0 1 1 0 t 7 0 1 1 1 t 8 1 0 0 0 t 9 1 0 0 1 t 10 1 0 1 0 t 11 1 0 1 1 t 1 1 1 0 0 t 1 1 1 0 1 t 1 1 1 1 0 T 15 1 1 1 1 <TTT;TT> <TT; T; T; T; T; > <TT; T> <TT; T> <TT; T; T; T; TT; TT> <T; ; ; > <; > <T> <T; > <T; ; ; ; ; > <TT; T; T> <T;, ; > <; > <T; > <; > <T> (*) <TT; T; T; T> <T; ; ; > < T; > <T; ; ; > <T; T; T> (*) < T; T; T > (**) NULEOTIDES (for DN) (T): THYMINE (): UNINE (): YTOSINE (): DENINE (Phe): Phenylalanine (Ser): Serine (Tyr): Tyrosine (ys): ysteíne (Leu): Leucine (Pro): Proline (lu): lutamine acid (Trp): Triptophan (His): Histidine (rg): rginine (Iso): Isoleucine (Thr): Threonine (ln): lutamine (sn): sparagine (Lys): Lysine (Met): Methionine (*) (Val): Valine (la): lanine (sp): spartic acid (ly): lycine (Initiators) (*) Punctuations (terminators) (**) M I N O I D S TBLE 10 From this TBLE 10, we have the following Numerical Transforms of the twenty different amino acids and punctuation (or, terminators), as a System of Systems of Boolean rithmetical Functions of nucleotides (for DN), except the correspondent to the TRYPTOPHN given by the numerical expression (7), because of its correspondence with the unique codon, T. Then, we have: 9

1) Determination of the Numerical Transform of the amino acid Phenylalanine for DN: 1111 1111 0000 0000 1111 1111 0000 0000 T T 1111 1111 0000 0000 (7) ( F) = NT { Phe} = NT 6x =.;T, { } or 1111 1111 0000 0000 T 1111 1111 0000 0000 1100 1100 1100 1100 8 8 70 (7) NT{} F = NT { Phe } =.{ ;T } ( 76) ( 10) ( ) (75) NT{} F = NT { Phe } = ( ( F) ( E) ) ( 1 0 ).{ ;T} 16 ) Determination of the Numerical Transform of the amino acid Serine (for DN) 1111 1111 0000 0000 1100 1100 1100 1100 T 1111 1111 0000 0000 1111 1111 0000 0000 1100 1100 1100 1100 1100 1100 1100 1100 1111 1111 0000 0000 1100 1100 1100 1100 1010 1010 1010 1010 (76) () S { } 18 x = NT Ser = NT =.{ ; T} 1111 1111 0000 0000 1100 1100 1100 1100 1111 0000 1111 0000 1010 1010 1010 1010 1111 0000 1111 0000 T 1111 1111 0000 0000 1010 1010 1010 1010 1111 0000 1111 0000 1100 1100 1100 1100 9

( 75) ( 0) ( 7) ( 0) ( 7 6 5 ) ( 1 0) (77) NT{} S = NT { Ser} =.{ ;T } 756 10 ( 7 ) ( 5 1) ( 6 ) ( 0) ( 7 6 5 1 0) 6 ( ) { } = NT { Ser} = ( FFFF ) ( FDDB) ( B E) ( ) ( FFD) ( FD89 ) ( B ) ( 908) ( 16F7 ) ( 1D ) ( 76 ) ( 5) ( 165 ) ( 181 ) ( ) ( 0 ) ].{ ;T} (78) NT S [ 7 95 16 ) Determination of the Numerical Transform of the amino acid Tyrosine (for DN) 1111 1111 0000 0000 1010 1010 1010 1010 T 1111 1111 0000 0000 (79) ( Y) = NT { Tyr} = NT 6x =.;T, { } or 1111 1111 0000 0000 1010 1010 1010 1010 1100 1100 1100 1100 (( 7 5) ) (( 0 ) ) (80) NT { Y } = NT { Tyr} =.{ ; T } ( 7 5 6 ) ( 1 0) ( ) (81) NT{ Y} = NT { Tyr} = ( F D E ) ( 1 1 1 0 ).{ ; T 16 } 9

) Determination of the Numerical Transform of the amino acid ysteine (for DN) 1111 1111 0000 0000 1111 0000 1111 0000 T 1111 1111 0000 0000 (8) ( ) = NT { ys} = NT 6x =.;T, { } or 1111 1111 0000 0000 1111 0000 1111 0000 1100 1100 1100 1100 750 (8) NT{} = NT { ys} =.{ ; T } 765 10 ( ) (8) NT{} = NT{ ys} = ( F)( E)( D)( ) ( D)( 1)()() 1 0.{ ; T 16 } 5) Determination of the Numerical Transform of the amino acid Leucine (for DN) 1100 1100 1100 1100 T 1111 1111 0000 0000 T 1111 1111 0000 0000 1100 1100 1100 1100 T 1111 1111 0000 0000 1100 1100 1100 1100 1100 1100 1100 1100 T 1111 1111 0000 0000 1010 1010 1010 1010 (85) L ( ) { } 18 x = NT Leu = NT =.{ ; T} 1100 1100 1100 1100 T 1111 1111 0000 00 00 1111 0000 1111 0000 1111 1111 0000 0000 T 1111 1111 0000 0000 1010 1010 1010 1010 1111 1111 0000 0000 T 1111 1111 0000 0000 1111 0000 1111 0000 95

( 7) ( 0) ( 7) ( 50) ( 7 6 ) ( 5 1 0) (86) NT{} L = NT { Leu} =.{ ;T } 76 510 ( 7 6) ( 1 0) 7610 6 ( ) (87) NT{} L = NT { Leu} = [ ( FFFF ) ( FDF7 ) ( 1 FF ) ( 1F7) ( FFBE) ( FDB6 ) ( 16 BE) ( 1B6) ( 5B9 ) ( 589 ) ( 9 ) ( 1) 5B08 5900 08 0 ]. ;T ( ) ( ) ( ) ( ) { } 6) Determination of the Numerical Transform of the amino acid Proline (for DN) 1100 1100 1100 1100 1100 1100 1100 1100 T 1111 1111 0000 0000 1100 1100 1100 110 0 1100 1100 1100 1100 1100 1100 1100 1100 (88) P ( ) { Pr o} 1 x = NT = NT = 1100 1100 1100 1100.{ ; T} 1100 1100 1100 1100 1010 1010 1010 1010 1100 1100 1100 1100 1100 1100 1100 1100 1111 0000 1111 0000 16 96

(71)(60) (70) (89) NT{} P = NT { Pr o } =.{ ;T } (7 6 1 0) (7 1 6 0 ) ( ) {} = NT { } = ( )( )( )( ) ( FFE ) ( FF6 ) ( 08 ) ( 1) ( DFF ) ( DF7 ) ( 9 ) ( 1) ( DFE ) ( DF6 ) ( 8 ) ( 0 ) ].{ ;T} (90) NT P Pr o [ FFF FF7 09 01 16 7) Determination of the Numerical Transform of the amino acid lutamine acid (for DN) 1111 0000 1111 0000 1010 1010 1010 1010 1010 1010 1010 1010 (91) E ( ) { } 6 x = NT lu = NT =.{ ; T} 1111 0000 1111 0000 1010 1010 1010 1010 1111 0000 1111 0000 (9) ( ) (7 ) ( 0) NT{} E = NT { lu} =.{ ;T } ( (7 5) ( 0) ) ( ) ( ) { } (9) NT{} E = NT { lu} = ( F 5)( 1 0 ). ; T 16 8) Determination of the Numerical Transform of the amino acid Triptophan (for DN) 97

(9) 1111 1111 0000 0000 ( W) = NT { Trp} = NT 1111 0000 1111 0000.{ ;T } x = 1111 0000 1111 0000 ' (95) NT{ W} = NT { Trp} = = 7 0. { ;T }, as we have seen in the expression (7). 9) Determination of the Numerical Transform of the amino acid Histidine (for DN) (96) ( ) { } H x 6 1100 1100 1100 1100 1010 1010 1010 1010 T 1111 1111 0000 0000 = NT His = NT = 1100 1100 1100 1100 1010 1010 1010 1010 1100 1100 1100 1100 { T}. ; (97) { } NT { His} NT H (7 5 1) (6 0) = =.{ ;T } ( 7 5 0) ( ) { } NT { His} ( ) ( ) { } (98) NT H = = F D 1 8 7 5 1 0. ; T 16 10) Determination of the Numerical Transform of the amino acid rginine (for DN) 98

( ) { } (99) R x 18 1100 1100 1100 1100 1111 0000 1111 0000 T 1111 1111 0000 0000 1100 1100 1100 1100 1111 0000 1111 0000 1100 1100 1100 1100 1100 1100 1100 1100 1111 0000 1111 0000 1010 1010 1010 1010 = NT rg = NT =.{ ; T} 1100 1100 1100 1100 1111 0000 1111 00 00 1111 0000 1111 0000 1010 1010 1010 1010 1111 0000 1111 0000 1010 1010 1010 1010 1010 1010 1010 1010 1111 0000 1111 0000 1111 0000 1111 0000 751 60 ( 750) ( 7 6 5 1 0) (100) NT{ R } = NT { rg} =. ( ) { ;T } 70 ( ( 7 ) ( 5 0 ) ) ( ( 7 ) ( 0 ) ) 6 ( ) (101) NT{ R } = NT { rg} = [ ( FFFF ) ( FDD7 ) ( 1FF ) ( 1DE) ( DB ) ( D900 ) ( 8 ) ( 8000) 7FFF 7DD 16FF 1D ( ) ( ) ( ) ( ) ( 5B ) ( 5900 ) ( ) ( 0 ) ].{ ;T} 16 99

11) Determination of the Numerical Transform of the amino acid Isoleucine (for DN) 1010 1010 1010 1010 T 1111 1111 0000 0000 T 1111 1111 0000 0000 1010 1010 1010 1010 (10) I () = NT { Iso} = NT T = 1111 1111 0000 0000 9x 1100 1100 1100 1100 1010 1010 1010 1010 T 1111 1111 0000 0000 1010 1010 1010 1010 { T}. ; (7 ) ( 0) (10) NT{} I = NT { Iso} = (7 6 ) (5 1 0). { ;T } (7 ) (5 0) ( ) {} = NT { Iso} = (( ) ( ) ( ) ( )) (( ) ( ) ( ) ( )) 16 { } (10) NT I [ 1FF D 1F7 D 1D 8 15 0 ]. ;T 1) Determination of the Numerical Transform of the amino acid Threonine (for DN) 1010 1010 1010 1010 1100 1100 1100 1100 T 1111 1111 0000 0000 1010 1010 1010 10 10 1100 1100 1100 1100 1100 1100 1100 1100 (105) T ( ) { Thr} 1 x = NT = NT = 1010 1010 1010 1010.{ ; T} 1100 1100 1100 1100 1010 1010 1010 1010 1010 1010 1010 1010 1100 1100 1100 1100 1111 0000 1111 0000 500

(7 5 1) (6 0) (7 0) (106) NT{} T = NT { Thr } =.{ ;T } (7 5 0) (7 5 1 6 0) ( ) {} = NT { Thr} = ( )( )( )( ) ( FFE ) ( 6D ) ( B ) ( 00) ( DFF ) ( D ) ( 9D ) ( 1) ( DFE ) ( D ) ( 9 ) ( 0 ) ].{ ;T} (107) NT T [ FFF 6D BD 01 16 1) Determination of the Numerical Transform of the amino acid lutamine (for DN) 1100 1100 1100 1100 1010 1010 1010 1010 1010 1010 1010 1010 (108) Q ( ) { ln} 6 x = NT = NT =.{ ; T} 1100 1100 1100 1100 1010 1010 1010 1010 1111 0000 1111 0000 (7 0) (109) NT{ Q } = NT { ln} =. { ;T } ( 7 5 1 6 0) { } NT { } ( ) { } (110) NT Q = ln = F 5 1B 1 E 1 0. ;T 16 ( ) 501

1) Determination of the Numerical Transform of the amino acid sparagine (for DN) 1010 1010 1010 1010 1010 1010 1010 1010 T 1111 1111 0000 0000 (111) ( N) { sn} 6 x = NT = NT = 1010 1010 1010 1010 1010 1010 1010 1010 1100 1100 1100 1100 (7 1) (6 0) (11) NT{ N } = NT { sn} =.{ ;T } ( 7 1 6 0) ( ) { T}. ; { } NT { } ( ) ( ) { } (11) NT N = N = F 9 E 1 7 1 6 0. ; T 16 15) Determination of the Numerical Transform of the amino acid Lysine (for DN) 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 (11) K ( ) { Lys} 6 x = NT = NT =.{ ; T} 1010 1010 1010 1010 1010 1010 1010 1010 1111 0000 1111 0000 8 (7 0) (115) NT{ K } = NT { Lys} =. { ;T } ( 7 1) ( 6 0) ( ) ( ) { } (116) NT{ K} = NT { Lys} = ( F 1) ( E 0 ). ; T 16 50

16) Determination of the Numerical Transform of the amino acid Methionine (It is also a Initiator) (for DN) (117) 1010 1010 1010 1010 ( M) = NT { Met} = 1111 1111 0000 0000 NT T.{ ;T } x = 1111 0000 1111 0000 (118) NT M Met 7 6 5 1 0. ; T { } = NT { } = T = ( ) ( ) ( ) ( ) { } ' 17) Determination of the Numerical Transform of the amino acid Valine (for DN) 1111 0000 1111 0000 1111 1111 0000 0000 T T 1111 1111 0000 0000 1111 0000 1111 00 00 T 1111 1111 0000 0000 1100 1100 1100 1100 (119) V ( ) { Val} 1 x = NT = NT = 1111 0000 1111 0000.{ ; T} T 1111 1111 0000 0000 1010 1010 1010 1010 1111 0000 1111 0000 T 1111 1111 0000 0000 1111 0000 1111 0000 7 0 76 5 10 (10) NT{ V} = NT { Val } =.{ ;T } (7 6) ( ) (5 ) (1 0) 7 5 0 ( ) 50

{ } = NT { } = ( ) ( ) ( ) ( ) ( 6D ) ( 6D ) ( 69 ) ( 69) ( B6D ) ( 965 ) ( 9D ) ( 95) ( 8 ) ( 0 ) ( 8 ) ( 0 ) ] 16. { ;T} (11) NT V Val [ FFF FF7 FBF FB7 18) Determination of the Numerical Transform of the amino acid lanine (for DN) 1111 0000 1111 0000 1100 1100 1100 1100 T 1111 1111 0000 0000 1111 0000 1111 00 00 1100 1100 1100 1100 1100 1100 1100 1100 (1) ( ) { la} 1 x = NT = NT = 1111 0000 1111 0000.{ ; T} 1100 1100 1100 1100 1010 1010 1010 1010 1111 0000 1111 0000 1100 1100 1100 1100 1111 0000 1111 0000 7 5 1 6 0 ( 7 0) (1) NT{ } = NT { la } =.{ ;T } (7 6 5 6 1 0) ( 7 5 0) ( ) { } = NT { } = ( ) ( ) ( ) ( ) ( 6D ) ( 6D ) ( 08 ) ( 00) ( DFF ) ( DF7 ) ( 9D ) ( 95) ( D ) ( D ) ( 8 ) ( 0 ) ] 16. { ;T} (1) NT la [ FFF FF7 BD B5 50

19) Determination of the Numerical Transform of the amino acid spartic acid (for DN) 1111 0000 1111 0000 1010 1010 1010 1010 T 1111 1111 0000 0000 (15) D ( ) { sp} 6 x = NT = NT =.{ ; T} 1111 0000 1111 0000 1010 1010 1010 1010 1100 1100 1100 1100 (7 5) ( 1) (6 ) ( 0) (16) NT{ D } = NT { sp} =.{ ;T } (7 5 6 1 0 ) ( ) (17) NT{ D} = NT { sp } = [( F )( D)( E)( ) ( 1B)( 9)( 1 )( 8) F 5 6 1 1 1 0 ]. ; T ( )( )( )( ) ( )( )( )( ) { } 0) Determination of the Numerical Transform of the amino acid lycine (for DN) 1111 0000 1111 0000 1111 0000 1111 0000 T 1111 1111 0000 0000 1111 0000 1111 00 00 1111 0000 1111 0000 1100 1100 1100 1100 (18) ( ) { ly} 1 x = NT = NT = 1111 0000 1111 0000.{ ; T} 1111 0000 1111 0000 1010 1010 1010 1010 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 16 505

7 1 6 0 ( 7 6 1 0) (19) NT{ } = NT { ly } =.{ ;T } ( (7 6) (1 0) ) ( 7 0) ( ) (10) NT{ } = NT { ly } = [ ( FFF ) ( FF7 ) ( FBF ) ( FB7) ( 8 ) ( 0 ) ( 08 ) ( 00) DF DF7 DBF DB7 1) The INITITOR ROUPS (for DN) ( ) ( ) ( ) ( ) ( 8 ) ( 0 ) ( 8 ) ( 0 ) ] 16. { ;T} There are three Initiator odons: Y 1,1, Y 1, and Y 1, : b) The first, Y 1,1, is the codon T, which have its Numerical Transform of Methionine amino acid, given by the numerical expressions, (11) or (1), as we have seen: 1010 1010 1010 1010 (11) ( Y1,1 ) = NT {( T(*) )} = NT T 1111 1111 0000 0000.{ ;T } x = 1111 0000 1111 0000 ' (1) NT {( T(*) )} = T = ( 7 ) ( 6 ) ( 5 1) ( 0 ).{ ;T } (b) The second, Y 1,, is the codon T, which it is a Valine codon when present at internal position. It have the following Numerical Transform: 1111 0000 1111 0000 (1) ( Y1, ) = NT {( T(*) )} = NT T 1111 1111 0000 0000.{ ;T } x = 1010 1010 1010 1010' (1) NT {( T(*) )} = T = ( 7 6) ( 6 ) ( 5 ) ( 1 0 ).{ ;T } 506

(c) The third, Y 1,, is the codon T, which it is also a Valine codon when present at internal position. It have the following Numerical Transform: 1111 0000 1111 0000 (15) ( Y1, ) = NT {( T(*) )} = 1111 1111 0000 0000 NT T =.{ ;T } x 1111 0000 1111 0000' (16) NT {( T(*) )} = T = 7 5 0. { ; T} ) The PUNTUTIONS (or, TERMINTOR ROUPS) (for DN) There are three Terminator odons: Y,1, Y, and Y, : (a) The first, Y 1,1, is the codon T ( ocre ), which it is not an amino acid and have the following Numerical Transform given by the numerical expressions, (17) or (18): 1111 1111 0000 0000 (17) ( Y,1 ) = NT {( T(**) )} = 1010 1010 1010 1010 NT =.{ ;T } x 1010 1010 1010 1010' (18) NT {( T(**) )} = =( 7 ) ( 0 ).{ ;T } (b) The second, Y,, is the codon T ( ambar ), which it is not an amino acid and have the following Numerical Transform given by the numerical expressions, (19) or (10): 1111 1111 0000 0000 (19) ( Y, ) = NT {( T(**) )} = NT 1010 1010 1010 1010 =.{ ;T } x 1111 0000 1111 0000 ' (10) NT {( T(**) )} = = ( 7 5) ( 6 ) ( 1) ( 0 ).{ ;T } (c) The third, Y,, is the codon T ( opal ), which it is not an amino acid and have the following Numerical Transform given by the numerical expressions, (11) or (1): 507

(11) 1111 1111 0000 0000 ( Y, ) = NT {( T (**))} = 1111 0000 1111 0000 NT =.{ ;T } x 1010 1010 1010 1010 (1) NT T (**) 7 6 5 1 0. ; T {( )} = = ( ) ( ) ( ) ( ) { } ' 508