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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 • • • • • • • 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 pk | a b | p pk 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 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 nm1 pm1 : 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 • • • • 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!