A mathematical model of the genetic code: structure and applicationsAntonino Sciarrino Universit di Napoli Federico II INFN, Sezione di Napoli

TAG 2006 Annecy-leVieux, 9 November 2006

Mathematical Model of the Genetic CodeWork in collaboration withLuc FRAPPATPaul SORBADiego COCURULLO

SUMMARYIntroductionDescription of the modelApplications : Codon usage frequencies DNA dimers free energyWork in progress

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

Gamow diamond codeGamow, Nature (1954)Nucleotides aredenoted by number 1,2,3,4 Amino-acids FIT the rhomb -shaped holes formed by the 4 nucleotides 20 a.a. !

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

The genetic code

Crystal basis model of the genetic code 4 basis C, U/T (Pyrimidines) G, A (Purines) are identified by a couple of spin labels (+ 1/2, - -1/2)L.Frappat, A. Sciarrino, P. Sorba: Phys.Lett. A (1998)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)

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 In each 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))

DINUCLEOTIDE Representation Content

Crystal basis model of the genetic codeCodons are composite states ( 64 basis vectors of (1/2, 1/2) )

belonging to sets identified by half- integer JH 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))

Codons in the crystal basis

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

Codon usage in Homo sap.

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 standardeukaryotic genetic code ascribable to its organisation and itsevolution

Let us define the codon usage probability for the codon XZN (X,Z,N {A,C,G,UT in DNA} )P(XZN) = limit n n XZN / N tot n XZN number of times codon XZN used in the processes N tot total number of codons in the same processes

For fixed XZ Normalization N P(XZN) = 1 Note - Sextets are considered quartets + doublets 8 quartets

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

Specimen (GenBank Release 149.0 09/2005 - Ncodons > 100.000) 26 VERTEBRATES 28 INVERTEBRATES 38 PLANTSTOTAL - 92 Biological species

Correlation coefficient VERTEBRATES

rXY

r CA

r UG

r UC

r AG

r UA

r CG

P

-0.89

-0.69

-0.75

-0.55

-0.76

-0.21

T

-0.92

-0.71

-0.89

-0.68

-0.91

-0.40

A

-0.88

-0.53

-0.89

-0.60

-0.76

-0.30

S

-0.92

-0.77

-0.87

-0.60

-0.75

-0.51

V

-0.84

-0.93

-0.69

-0.74

-0.68

-0.53

L

-0.83

-0.93

-0.87

-0.91

-0.87

-0.69

R

-0.90

-0.93

-0.39

-0.27

-0.41

-0.11

G

-0.94

-0.89

-0.75

-0.74

-0.77

-0.56

a.a.

-0.89

-0.80

-0.76

-0.64

-0.74

-0.41

Correlation coefficient PLANTS

rXY

r CA

r UG

r UC

r AG

r UA

r CG

P

-0.91

-0.81

-0.54

-0.61

-0.41

-0.48

T

-0.94

-0.87

-0.79

-0.59

-0.75

-0.48

A

-0.94

-0.93

-0.72

-0.57

-0.63

-0.55

S

-0.87

-0.86

-0.75

-0.78

-0.71

-0.56

V

-0.66

-0.72

-0.75

-0.65

-0.71

-0.15

L

-0.72

-0.85

-0.57

-0.52

-0.54

-0.17

R

-0.76

-0.66

-0.67

-0.50

-0.16

-0.49

G

-0.83

-0.48

-0.73

-0.14

-0.36

-0.07

a.a.

-0.83

-0.77

-0.69

-0.55

-0.53

-0.37

Correlation coefficient INVERTEBRATES

rXY

r CA

r UG

r UC

r AG

r UA

r CG

P

-0.78

-0.63

-0.50

-0.74

-0.20

-0.52

T

-0.85

-0.87

-0.74

-0.76

-0.62

-0.60

A

-0.82

-0.79

-0.75

-0.68

-0.51

-0.53

S

-0.91

-0.83

-0.71

-0.86

-0.55

-0.79

V

-0.78

-0.92

-0.66

-0.78

-0.72

-0.46

L

-0.49

-0.92

-0.48

-0.66

-0.50

-0.25

R

-0.55

-0.76

-0.76

-0.27

-0.01

-0.53

G

-0.73

-0.48

-0.57

-0.14

-0.02

-0.08

a.a.

-0.74

-0.78

-0.65

-0.61

-0.38

-0.47

Averaged value of P(..N)

Averaged value of P(..N)

Averaged value of sum of two correlated P(N)

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

Indication for correlation for codon usage probabilities P(A) and P(C) ( P(U) and P(G)) for quartets.

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

Eucar.

r XZA-XZA

r XZC-XZ C

r XZG-XZG

r XZ U-XZ U

Ser-Thr

0.88

0.94

0.90

0.80

Ser-Pro

0.93

0.90

0.87

0.91

Ser-Ala

0.86

0.93

0.82

0.81

Thr-Pro

0.83

0.91

0.93

0.74

Thr-Ala

0.91

0.93

0.93

0.94

Pro-Ala

0.86

0.93

0.93

0.94

Leu-Val

0.85

0.82

-0.70

0.96

The set of 8 quartets splits into 3 subsets4 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)

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

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

ASSUMPTION

SUM RULESK INDEPENDENT OF THE b.s.XZ QUARTETS

SUM RULES Theoretical correlation matrixXZ = NC,CG,GG,CU,GU

Observed averaged value of the correlation matrix , in red thetheoretical value

Irrep. JH, JV

Codons

3/2, 3/2

Pro CCC, Ser UCC, Ala GCC, Thr ACC

(1/2, 3/2) 1

Pro CCU, Ser UCU, Ala GCU, Thr ACU

(3/2, 1/2) 1

Pro CCG, Ser UCG, Ala GCG, Thr ACG

(1/2, 1/2) 1

Pro CCA, Ser UCA, Ala GCA, Thr ACA

(1/2, 3/2) 2

Leu CUC, Leu CUU, Val GUC, Val GUU

(1/2, 1/2) 2

Leu CUG, Leu CUA, Val GUG, Val GUA

Shannon EntropyLet us define the Shannon entropy for the amino-acid specified by the first two nucleotide XZ (8 quartes)

Shannon EntropyUsing the previous expression for P(XZN) we get

N (XZN), HbsN Hbs(XZN), PN P(XZN) SXZ largely independent of the b.sp.

Shannon Entropy

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

DINUCLEOTIDE Representation Content

SUM RULES for FREE ENERGY

Comparison with exp. dataG in Kcal/mol

DINUCLEOTIDE Distribution

Comparison with experimental data

Work in progress and future perspectivesFron 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)N of U q 0 (sl(2)H sl(2)V) New pametrization of nucleotidees sequences

Spin parametrisation

Algorithm for the spin parametrisation of orderedn-nucleotide sequence

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

For each gene of Homo Sap. (total ~28.000 genes)Consider the N-nucleotide coding sequence (CDS)Compute the labels JH, J3H ; JV, J3V for any n-nucleotide subsequence (1 n N) Plot labels versus n

Red JH - Green J3H Blue JV - Black J3V

Red JH - Green J3H Blue JV - Black J3V

Red JH - Green J3H Blue JV - Black J3V

Red JH - Green J3H Blue JV - Black J3V

Numerical estimatorDefine for any sequence of length NPlot number of CDS with the same value of Diff (Sum) versus Diff (Sum) Compute Diff (Sum) for 28.000 random sequences (300 < N < 4300)with uniform probability for each nucleotideComparison number of CDS - random sequences

ConclusionsCorrelations in codon usage frequencies computed over the whole exonic region fit well in the mathematical scheme of the crystal basis model of the genetic code Missing explanation for the correlations Formalism of crystal basis model useful to parametrize free energy for DNA dimersMore generally, use of U q 0 (sl(2)H sl(2)V) mathematical structure may be useful to describe sequences of nucleotides .