p-ADIC APPROACH TO THE GENETIC CODE AND THE GENOME

1. p-ADIC APPROACH TO THE GENETIC CODE AND THE GENOME Branko Dragovich Institute of Physics, Belgrade, Serbia TAG, 20 – 24 Oct 2008, Annecy

2. 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

3. 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

4. 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.

5. DNA and RNA

7. Nucleotides (bases) and codons Nucleotides: C, A, U (T), G Codons: ordered trinucleotides 4 x 4 x 4 = 64 codons

8. Amino acids

9. (Universal) Genetic Code

10. 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

11. p-adic codon space: p-Adic Modeling of the Genetic Code C (cytosine) = 1, A (adenine) = 2, T (thymine) = U (uracil) = 3, G (guanine) = 4 ( 0 = absence of nucleotide )

12. Our Table of Vertebral Mitochondrial Code

13. 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

14. 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

15. p-Adic Genomic Space where Definition: (p, q)-adic genomic space is a double is a set of natural numbers, and is q-adic distance.

16. Examples of p-Adic Genomic Spaces • 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

17. 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)

18. 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!