p-ADIC APPROACH TO

THE GENETIC CODE AND

THE GENOME

Branko Dragovich Institute of Physics, Belgrade, Serbia

TAG, 20 – 24 Oct 2008, Annecy

• 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 | | k

pn p

| | kpa b p

| | | | 1pp

n n

INTRODUCTION

p-ADIC MATHEMATICAL PHYSICS

• 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

• 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

20 1 2[64] { 5 5 : 1,2,3,4}iC n n n n

p-adic codon space:

C (cytosine) = 1, A (adenine) = 2, T (thymine) = U (uracil) = 3, G (guanine) = 4 ( 0 = absence of nucleotide )

20 1 2 0 1 25 5n n n n n n

Our Table of Vertebral Mitochondrial Code

111 CCC Pro 112 CCA Pro 113 CCU Pro 114 CCG Pro

211 ACC Thr 212 ACA Thr 213 ACU Thr 214 ACG Thr

311 UCC Ser 312 UCA Ser 313 UCU Ser 314 UCG Ser

411 GCC Ala 412 GCA Ala 413 GCU Ala 414 GCG Ala

121 CAC His 122 CAA Gln 123 CAU His 124 CAG Gln

221 AAC Asn 222 AAA Lys 223 AAU Asn 224 AAG Lys

321 UAC Tyr 322 UAA Ter 323 UAU Tyr 324 UAG Ter

421 GAC Asp 422 GAA Glu 423 GAU Asp 424 GAG Glu

131 CUC Leu 132 CUA Leu 133 CUU Leu 134 CUG Leu

231 AUC Ile 232 AUA Met 233 AUU Ile 234 AUG Met

331 UUC Phe 332 UUA Leu 33 3 UUU Phe 334 UUG Leu

431 GUC Val 432 GUA Val 433 GUU Val 434 GUG Val

141 CGC Arg 142 CGA Arg 143 CGU Arg 144 CGG Arg

241 AGC Ser 242 AGA Ter 243 AGU Ser 244 AGG Ter

341 UGC Cys 342 UGA Trp 343 UGU Cys 344 UGG Trp

441 GGC Gly 442 GGA Gly 443 GGU Gly 444 GGG Gly

p-Adic Properties of the Vertebral Mitochondrial Code

• T-symmetry: doublets-doublets and quadruplets-quadruplets 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

( [( 1) ], )mp qp d

where 1

0 1 1[( 1) ] { : 1,2, , 1; }m mp m ip n n p n p n p m

qd

is a set of natural numbers, and

is q-adic distance.

Examples of p-Adic Genomic Spaces

[( 1) ]mp p

• 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

• cDNA and RNA space: p = 61• ncDNA space: p = 5 • Protein space: p = 23

(20 standard + selenocysteine + pyrrolysine)

2

1

[ ] [( 1) ]m m

p pm mB N p

Conclusion

• 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!