Theoretical and Mathematical Physics, Vol. 128, No. 1, pp. 856–869, 2001 QUANTUM GROUPS AND THE GENETIC CODE L. Frappat, 1 P. Sorba, 1 and A. Sciarrino 2 The quantum algebra Uq (sl(2) ⊕ sl(2)) in the limit q → 0 is proposed as a symmetry algebra for the genetic code. The nucleotide triplets (codons) in the DNA chain are classified in crystal bases. This construction can be compared to the baryon classification from quarks in particle physics, one main difference being the natural order in the state constituents provided by the crystal base. An operator ensuring the correspondence between codons and amino acids is constructed from the above algebra: two codons corresponding to the same or different amino acids acquire the same or different eigenvalues respectively. Then a set of relations between the physicochemical properties of the amino acids are derived and compared with the experimental data. Correlations of the codon usage for quartets and sextets are determined, fitting naturally in the framework of this model. 1. Introduction The perfect correspondence between nucleotide triplets in the desoxyribonucleic acid (DNA) sequence and the amino acids is called the genetic code [1]. The DNA macromolecule, which controls the synthesis of proteins (the most abundant organic substances in living systems), is made of two linear chains of nucleotides in the famous double helix structure. Each nucleotide is characterized by one of the four elementary bases: adenine (A) and guanine (G) derived from purine and cytosine (C) and thymine (T) derived from pyrimidine. The DNA is localized in the cell nucleus, and messenger ribonucleic acid (mRNA) carries the genetic information to the cytoplasm. This operation of information transfer from DNA to mRNA is called transcription, the DNA bases A, G, C, and T being respectively associated to the mRNA bases U, C, G, and A (U is the uracile base). A codon is an ordered sequence of three bases, and there are 4×4×4 different codons. In the genetic code, each of the 64 codons either is related through a ribosome to an amino acid or is used as a termination signal. The latter codons are called nonsense or stop codons, and their role is to stop the biosynthesis. Twenty different amino acids occur in the proteins in living systems. Therefore, different codons can be related to the same amino acid: the genetic code is degenerate. In the standard eukaryotic code (Table 1), the codons are organized in multiplets, each one corresponding to a specific amino acid. 2. The model [2] We consider the four nucleotides as basic states of the (1/2, 1/2) representation of the quantum algebra U q (sl(2) ⊕ sl(2)) in the limit q → 0. Constructing the tensor product of three such four-dimensional representations, we obtain a codon. Actually, this approach mimics the group theory classification of baryons made of three quarks in particle physics, the building blocks here being the A, C, G, and T (U) nucleotides. The essential difference is that a codon is an ordered set of three nucleotides, which is not the case for a baryon: there are three different codons made of the A, A, and U nucleotides (namely, 1 Laboratoire d’Annecy-le-Vieux de Physique Th´ eorique LAPTH CNRS, UMR 5108, associ´ ee `a l’Universit´ e de Savoie BP 110, F-74941 Annecy-le-Vieux CEDEX, France. 2 Dipartimento di Scienze Fisiche, Universit`a di Napoli Federico II; I.N.F.N., Sezione di Napoli, Italy; Complesso di Monte, S. Angelo, Via Cintia, I-80126 Napoli, Italy. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 128, No. 1, pp. 27–42, July, 2001. 856 0040-5779/01/1281-0856$25.00 c  2001 Plenum Publishing Corporation