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A mathematical model of the genetic code: structure and applications

A mathematical model of the genetic code: structure and applications. Antonino Sciarrino Università di Napoli “Federico II” INFN, Sezione di Napoli TAG 2006 Annecy-leVieux, 9 November 2006.

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A mathematical model of the genetic code: structure and applications

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  1. A mathematical model of the genetic code: structure and applications Antonino Sciarrino Università di Napoli “Federico II” INFN, Sezione di Napoli TAG 2006 Annecy-leVieux, 9 November 2006

  2. Mathematical Model of the Genetic Code Work in collaboration with Luc FRAPPAT Paul SORBA Diego COCURULLO

  3. SUMMARY • Introduction • Description of the model • Applications : Codon usage frequencies DNA dimers free energy • Work in progress

  4. It is amazing that the complex biochemical relations between DNA and proteins were very quickly reduced to a mathematical model. Just few months after the WATSON-CRICK discovery G. GAMOW proposed the “diamond code”

  5. Gamow “diamond code” Gamow, Nature (1954) Nucleotides are denoted by number 1,2,3,4 Amino-acids FIT the rhomb -shaped “holes” formed by the 4 nucleotides  20 a.a. !

  6. Since 1954 many mathematical modelisations of the genetic coded have been proposed (based on informatiom, thermodynamic, symmetry, topology… arguments) Weak point of the models: often poor explanatory and/or predictive power

  7. The genetic code

  8. Crystal basis model of the genetic code L.Frappat, A. Sciarrino, P. Sorba: Phys.Lett. A (1998) 4 basisC, U/T (Pyrimidines) G, A (Purines) are identified by a couple of “spin” labels (+  1/2, -  -1/2) Mathematically - C,U/T,G,A transform as the 4 basis vectors of irrep. (1/2, 1/2) of U q  0 (sl(2)H sl(2)V)

  9. Crystal basis model of the genetic code • Dinucleotides are composite states ( 16 basis vectors of (1/2, 1/2)2 ) belonging to “sets” identified by two integer numbers JH JV Ineach “set” the dinucleotide is identified by two labels - JH  JH,3  JH - JV  JV,3  JV Ex. CU = (+,+)  (+, -) ( JH = 1/2, JH,3 = 1/2; JV = 1/2, JV,3 = 1/2) Follows from property of U(q  0)(sl(2))

  10. DINUCLEOTIDE Representation Content

  11. Crystal basis model of the genetic code • Codons are composite states ( 64 basis vectors of (1/2, 1/2) ) belonging to “sets” identified by half- integerJH JV (“set”  irreducible representation = irrep.) Ex. CUA = (+,+)  (-, +)  (-,-) ( JH = 1/2, JH,3 = 1/2; JV = 1/2, JV,3 = 1/2) Follows from property of U(q  0)(sl(2))

  12. Codons in the crystal basis

  13. Codon usage frequency • Synonymous codons are not used uniformly (codon bias) • codon bias (not fully understood) ascribed to evolutive-selective effects • codon bias depends  Biological species (b.sp.)  Sequence analysed  Amino acid (a.a.) encoded  Structure of the considered multiplet  Nature of codon XYZ  …………………….

  14. Codon usage in Homo sap.

  15. Our analysis deals with global codon usage , i.e. computed over all the coding sequences (exonic region) for the b.sp. of the considered specimen  To put into evidence possible general features of the standard eukaryotic genetic code ascribable to its organisation and its evolution

  16. Let us define the codon usage probability for the codon XZN (X,Z,N  {A,C,G,UT in DNA} )P(XZN) = limit n   n XZN / N totn XZNnumber of times codon XZN used in the processes N tot total number of codons in the same processes For fixed XZ Normalization ∑NP(XZN) = 1 Note - Sextets are considered quartets + doublets  8 quartets

  17. Def. - Correlation coefficient rXY for two variables X P..XY P..Y

  18. Specimen (GenBank Release 149.0 09/2005 - Ncodons > 100.000) • 26 VERTEBRATES • 28 INVERTEBRATES • 38 PLANTS • TOTAL - 92 Biological species

  19. Correlation coefficient VERTEBRATES

  20. Correlation coefficient PLANTS

  21. Correlation coefficient INVERTEBRATES

  22. Averaged value of P(..N)

  23. Averaged value of P(..N)

  24. Averaged value of sum of two correlated P(N)  

  25. Ratios of obs2(X+Y) and th2(X+Y) = obs2(X)+ obs2(Y) averaged over the 8 a.a. for the sum of two codon probabilities

  26. Indication for correlation for codon usage probabilitiesP(A) and P(C) (P(U) and P(G)) for quartets.

  27. Correlation between codon probabilities for different a.a. • Correlation coefficients between the 28 couples P XZN-X’Z’N where XZ(X’Z’) specify 8 quartets. The following pattern comes out for the whole eucaryotes specimen (n = 92)

  28. The set of 8 quartets splits into 3 subsets • 4 a.a. with correlated codon usage (Ser, Pro, Arg, Thr) • 2 a.a. with correlated codon usage (Leu, Val) • 2 a.a. with generally uncorrelated codon usage (Arg, Gly)

  29. Statistical analysis   Correlation for P(XZA)-P(XZC),XZ quartets  Correlation for P(N) between {Ser, Pro, Thr, Ala} and {Leu, Val} The observed correlations well fit in the mathematical scheme of the crystal basis model of the genetic code

  30. In the crystal basis model P(XYZ) can be written as function of

  31. ASSUMPTION

  32. SUM RULES K INDEPENDENT OF THE b.s. XZ  QUARTETS

  33. SUM RULES  “Theoretical” correlation matrixXZ = NC,CG,GG,CU,GU

  34. Observed averaged value of the correlation matrix , in red the theoretical value

  35. Shannon Entropy Let us define the Shannon entropy for the amino-acid specified by the first two nucleotide XZ (8 quartes)

  36. Shannon Entropy Using the previous expression forP(XZN) we get N  (XZN), HbsN Hbs(XZN),PN  P(XZN)  SXZlargely independent of the b.sp.

  37. Shannon Entropy

  38. DNA dinucleotide free energy Free energy for a pair of nucleotides, ex. GC, lying on one strand of DNA, coupled with complementary pair, CG, on the other strand. CG from 5’  3’ correlated with GC from 3’  5’

  39. DINUCLEOTIDE Representation Content

  40. SUM RULES for FREE ENERGY

  41. Comparison with exp. data G in Kcal/mol

  42. DINUCLEOTIDE Distribution

  43. Comparison with experimental data

  44. Work in progress and future perspectives Fron the correspondence {C,U/T,G,A} I.R. (1/2,1/2) of U q  0 (sl(2)H sl(2)V)  Any ordered N nucleotides sequence  Vector of I.R.  (1/2,1/2)Nof U q  0 (sl(2)H sl(2)V)  New pametrization of nucleotidees sequences

  45. “Spin” parametrisation

  46. Algorithm for the “spin” parametrisation of orderedn-nucleotide sequence

  47. From this parametrisation: • Alternative construction of mutation model, where mutation intensitydoes not depend from the Hamming distance between the sequences, but from the change of “labels” of the “sets”. C. Minichini, A.S., Biosystems (2006) • Characterization of particular sequences (exons, introns, promoter, 5’ or 3’ UTR sequences,….) L. Frappat, P. Sorba, A.S., L. Vuillon, in progress

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