Algebraic Evolution of the Genetic Code

The evolution of the genetic code has been speculated by many authors, but all of them lack determinative proofs. Jukes [4] inferred the evolution from mitochondrial codes, which are different from the universal codes in some respects. He postulated an archetypal code containing 14 amino acids, from which the universal code could evolve by gene duplication followed by mutational changes. This process could have included 28 amino acids in the code, but it has only 20 ones due to the freezing, which is defined as some codons that could have coded for different amino acids code for the same one. Hornos et al. [3] devised an algebraic model for the evolution of the genetic code. They looked for Lie groups that have a 64-dimensional irreducible representation, which is called a codon representation. Especially they studied Sp(6), the symplectic group of degree 6. If the codon representation of Sp(6) is restricted to its subgroup Sp(4)*SU(2), then it is decomposed into 6 irreducible representations, which is called symmetry breaking. Each representation corresponds to one of the primordial amino acids and the termination codon. Decomposing the representation according to the chain of 4 subgroups, they obtained 27 subspaces assigned to the amino acids and the termination codon. As there are only 20 amino acids in the code, 6 spaces are redundant, which is freezing. Forger and Sachse [1] explored the codon representations of Lie superalgebras. If the codon representation of the orthosymplectic algebra osp(5|2) is restricted to its subalgebra sp(2)+so(5), then it is decomposed into 3 irreducible representations. Continuing the decomposition according to the chain of subalgebras, they obtained 26 subspaces. But they did not explain the mechanism of freezing, and did not assign amino acids to the subspaces. We assign the amino acids to the 26 subspaces of [1], and try to explain the mechanism of freezing.