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p-ADIC APPROACH
TO
THE GENETIC CODE
AND
THE GENOME
Branko Dragovich
Institute of Physics, Belgrade, Serbia
TAG, 20 – 24 Oct 2008, Annecy
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Introduction
p-Adic modeling of the genetic code
p-Adic vertebral mitochondrial code
p-Adic degeneracy of the genetic code
p-Adic genomic spaces
Evolution of the genetic code
Conclusion
INTRODUCTION
Question: p-adic ?
Example: 1, 2, 3, 4, 5, 6, 7, 8, 9, …
abs. value
|n| = n
conventional distance |a -b|
p-adic abs. value
p-adic distance
| n | | n | p  1
p
| n | p  pk
| a  b | p  pk
INTRODUCTION
p-ADIC MATHEMATICAL PHYSICS
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p-Adic numbers:
discovered by K. Hansel in 1897.
many applications in mathematics, e.g. representation theory, algebraic geometry
and modern number theory
many applications in mathematical physics since 1987, e.g. string theory, QFT,
quantum mechanics, dynamical systems, complex systems, ....
Third International Conference on p-Adic Mathematical Physics: from Planck
scale physics to complex systems to biology. (Steklov Mathem. Institute, Moscow,
1-6 Oct. 2007).
New intern. multidisciplinary journal: “p-Adic Numbers, Ultrametric Analysis,
and Applications”
Any p-adic number has a unique canonical representation
Real and p-adic numbers unify into adeles.
DNA and RNA
Nucleotides (bases) and codons
Nucleotides: C, A, U (T), G
Codons: ordered trinucleotides
4 x 4 x 4 = 64 codons
Amino acids
(Universal) Genetic Code
Modeling of the Genetic Code
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Gamov (1954), Crick (1957)
Rumer (1966), Crick, …
Jukes, Woese, Swanson, …
J. Hornos and Y. Hornos (1993), Forger and
Sachse (2000)
• Frappat, Sciarrino and Sorba (1998)
• p-Adic approach: B. Dragovich and A. Dragovich
(2006), Khrennikov and Kozyrev, Bradley
p-Adic Modeling of the Genetic Code
p-adic codon space:
C[64]  {n0  n1 5  n2 52 : ni  1, 2, 3, 4}
C (cytosine) = 1, A (adenine) = 2,
T (thymine) = U (uracil) = 3, G (guanine) = 4
( 0 = absence of nucleotide )
n0  n1 5  n2 52  n0 n1n2
Our Table of Vertebral Mitochondrial Code
111
112
113
114
121
122
123
124
131
132
133
134
141
142
143
144
CCC
CCA
CCU
CCG
CAC
CAA
CAU
CAG
CUC
CUA
CUU
CUG
CGC
CGA
CGU
CGG
Pro
Pro
Pro
Pro
His
Gln
His
Gln
Leu
Leu
Leu
Leu
Arg
Arg
Arg
Arg
211
212
213
214
221
222
223
224
231
232
233
234
241
242
243
244
ACC
ACA
ACU
ACG
AAC
AAA
AAU
AAG
AUC
AUA
AUU
AUG
AGC
AGA
AGU
AGG
Thr
Thr
Thr
Thr
Asn
Lys
Asn
Lys
Ile
Met
Ile
Met
Ser
Ter
Ser
Ter
311 UCC
312 UCA
313 UCU
314 UCG
321 UAC
322 UAA
323 UAU
324 UAG
331 UUC
332 UUA
33 3 UUU
334 UUG
341 UGC
342 UGA
343 UGU
344 UGG
Ser
Ser
Ser
Ser
Tyr
Ter
Tyr
Ter
Phe
Leu
Phe
Leu
Cys
Trp
Cys
Trp
411
412
413
414
421
422
423
424
431
432
433
434
441
442
443
444
GCC
GCA
GCU
GCG
GAC
GAA
GAU
GAG
GUC
GUA
GUU
GUG
GGC
GGA
GGU
GGG
Ala
Ala
Ala
Ala
Asp
Glu
Asp
Glu
Val
Val
Val
Val
Gly
Gly
Gly
Gly
p-Adic Properties of the Vertebral Mitochondrial Code
• T-symmetry: doublets-doublets and quadrupletsquadruplets invariance
• 5-Adic distance gives quadruplets
• 2-Adic distance inside quadruplets gives doublets
• Degeneration of the genetic code has p-adic structure
• p-Adic degeneracy principle: Codons code amino acids
and stop signals by doublets which are result of
combined 5-adic and 2-adic distances
Evolution of the genetic code
• Modern assignment of codon doublets to particular amino
acids may be a result of coevolution of the genetic code
and amino acids: single nucleotide code – 4 amino acids,
dinucleotide code – 16 amino acids, trinucleotide code
20+2 amino acids (selenocysteine, pyrrolysine).
• Other (15) codes may be regarded as slight modifications
of the Vertebral Mitochondrial Code
p-Adic Genomic Space
Definition: (p, q)-adic genomic space is a double
( p [( p 1) ], dq )
m
where
 p [( p 1)m ]  {n0  n1 p  nm1 pm1 : ni  1, 2, , p 1; m  }
is a set of natural numbers, and
dq
is q-adic distance.
Examples of p-Adic Genomic Spaces
 p [( p 1) ]
m
• 1-nucleotide codon space: p=5, m =1
• 2-nucleotide codon space: p=5, m =2
• 3-nucleotide codon space: p=5, m =3
• present space of amino acids: p=23, m=1
• possible (Jukes) space of amino acids:
p=29, m=1
• first space of amino acids: p=5, m=1
• second space of amino acids: p=17, m=1
p-Adic DNA, RNA and protein spaces
Bp [ N ] 
m2
m

[(
p

1)
]
p
m m
1
• cDNA and RNA space: p = 61
• ncDNA space: p = 5
• Protein space: p = 23
(20 standard + selenocysteine + pyrrolysine)
Conclusion
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Space of nucleotides is p-adic
Space of codons is p-adic
Genetic code has p-adic degeneration
Genomic spaces (spaces of nucleotides, codons, DNA,
RNA, amino acids, proteins) have p-1 structural units,
where p = prime number.
There is p-adics in genomics!