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A mathematical model of the genetic code: structure and applications

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### 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

Mathematical Model of the Genetic Code

Work in collaboration with

Luc FRAPPAT

Paul SORBA

Diego COCURULLO

SUMMARY

- Introduction
- Description of the model
- Applications : Codon usage frequencies

DNA dimers free energy

- Work 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 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. !

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

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)

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))

DINUCLEOTIDE Representation Content

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))

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

…………………….

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

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 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

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

Specimen (GenBank Release 149.0 09/2005 - Ncodons > 100.000)

- 26 VERTEBRATES
- 28 INVERTEBRATES
- 38 PLANTS
- TOTAL - 92 Biological species

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 probabilitiesP(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-X’Z’N where XZ(X’Z’) specify 8 quartets. The following pattern comes out for the whole eucaryotes specimen (n = 92)

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)

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

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

Shannon Entropy

Let us define the Shannon entropy for the amino-acid

specified by the first two nucleotide XZ (8 quartes)

Shannon Entropy

Using the previous expression forP(XZN) we get

N (XZN), HbsN Hbs(XZN),PN P(XZN)

SXZlargely independent of the b.sp.

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

Comparison with exp. data

G in Kcal/mol

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

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

Numerical estimator

Define for any sequence of length N

Plot 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 nucleotide

Comparison number of CDS -random sequences

Conclusions

- Correlations 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 dimers
- More generally, use of U q 0 (sl(2)H sl(2)V) mathematical structure may be useful to describe sequences of nucleotides .

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