As shown before, we have up to now uncovered a part of the palindromic degeneracy preserving symmetry. Observing Tables 15 and 18, is easily to understand its meaning as a transformation inside the genetic code connecting quartets with the same degeneracy distribution. On the mathematical-model side

Table 19 Palindromic symmetry; a degeneracy preserving transformation connecting different quartets of the genetic code

Table 19 Palindromic symmetry; a degeneracy preserving transformation connecting different quartets of the genetic code this symmetry is concisely represented by a simple operation: the complement to 1 of all the binary digits of a given string.

For completing the palindromic symmetry we need to place the remaining amino acids (four of degeneracy 4 and seven of degeneracy 2, plus the pair of codons representing the stop signal). This task is relatively easy because: (i) all the remaining degeneracy-4 aminoacids have a middle letter C while all the degeneracy-2 ones have a middle letter A; (ii) because on the model side there do not exist palindromic pairs of strings pertaining to a same number, there do not exist palindromic transformations inside a given quartet and thus a change in one or more letters of the codon, implying a quartet change, is needed. The most natural choice is to preserve the transformations uncovered until now but maintaining fixed the middle base.

By this choice we are able to place the amino acids Ser4, Thr, Tyr, Stop, Asn, and Lys; the associated transformation is represented by a exchange of U and A in the first letter of the codons. For the remaining amino acids, because of property (i) the unique possible choice is exchange of C and G letters in the first position of the codon. The complete symmetry is shown in Table 19.

Of course, at the end of this process, some remaining degeneracy still exists. The fundamental one is an indeterminacy with regard to the assignation of binary strings to quartets: the full set of strings assigned to a quartet can be exchanged with the full set of palindromic strings pertaining to a palindromic quartet. We shall not discuss here some numerical arguments for the final placements that we show in Table 18 representing our mathematical model of the genetic code. This is the most probable assignation taking into account our knowledge about the symmetry properties of the genetic code on one side, and the number representation system, on the other. As mentioned above, some room for ambiguity in this assignation still remains, but the main properties of the code are described and some additional uncovered symmetries are evidenced3 (see also the following Section 7). The palindromic symmetry can be considered as the counterpart of the first discovered

3 Some differences with the placement shown by Gonzalez (2004) are due to optimization regarding number theoretical properties that are roughly discussed here. All the differences (mainly inversion of palindromic amino acid pairs and variable ending assignation of Y-ending codons) are compatible with the degrees of freedom still remaining after codon assignation following the description in Sections 5 and 6, that is, observing the symmetry of the palindromic transformation and the other internal symmetries of the genetic code.

regularity in the degeneracy distribution of the genetic code: Rumer's transformation. The Russian theoretical physicist Y. Rumer (1966) observed that the genetic code could be divided into two halves, as represented in Table 20. One half corresponds to quartets of degeneracy 4 and the other to quartets containing degeneracies 3, 2, or 1. Rumer found that a global anti-symmetric transformation connects these two halves of the code. This transformation is represented by the following exchange between all the letters of a codon: U,C,A,G oG,A,C,U. We examine below the mathematical meaning of this transformation and its connection with the new concept of codon parity.

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