It is very surprising that Rumer's classification can also be presented in a completely analogous way. In Table 20 we show Rumer's classes, that is, the quartets of degeneracy 4 (given in grey) and the quartets of degeneracy-3, 2, and 1. It is easy

U |
C |
A |
G | ||

U |
TTT Phe |
TCT Ser |
TAT Tyr |
TGT Cys |
U |

TTC Phe |
TCC Ser |
TAC Tyr |
TGC Cys |
C | |

TTA Leu |
TCA Ser |
TAA Stop |
TGA Cys |
A | |

TTG Leu |
TCG Ser |
TAG Stop |
TGG Trp |
G | |

C |
CTT Leu |
CCT Pro |
CAT His |
CGT Arg |
U |

CTC Leu |
CCC Pro |
CAC His |
CGC Arg |
C | |

CTA Leu |
CCA Pro |
CAA Gln |
CGA Arg |
A | |

CTG Leu |
CCG Pro |
CAG Gln |
CGG Arg |
G | |

A |
ATT Ile |
ACT Thr |
AAT Asn |
AGT Ser |
U |

ATC Ile |
ACC Thr |
AAC Asn |
AGC Ser |
C | |

ATA Ile |
ACA Thr |
AAA Lys |
AGA Arg |
A | |

ATG Met |
ACG Thr |
AAG Lys |
AGG Arg |
G | |

G |
GTT Val |
GCT Ala |
GAT Asp |
GGT Gly |
U |

GTC Val |
GCC Ala |
GAC Asp |
GGC Gly |
C | |

GTA Val |
GCA Ala |
GAA Glu |
GGA Gly |
A | |

GTG Val |
GCG Ala |
GAG Glu |
GGG Gly |
G |

Fig. 3 Non-linear algorithmic representation of Rumer's class. Observe that the structure is completely analogous to that of parity class but acting on a window defined by the two first letters of the codon. The induced chemical dichotomies are K-Am in the middle letter (shared with the parity class algorithm) and W-S in the first one

Fig. 3 Non-linear algorithmic representation of Rumer's class. Observe that the structure is completely analogous to that of parity class but acting on a window defined by the two first letters of the codon. The induced chemical dichotomies are K-Am in the middle letter (shared with the parity class algorithm) and W-S in the first one to see that a non-linear algorithm with the same structure as the one for parity determination (Fig. 2) can be defined. Moreover, the analysis algorithm acts also on a window of two bases - the first two - and maintains the chemical dichotomy induced by parity determination in the middle base.

In fact, the middle base is shared by the two algorithms and thus, the right dichotomy class should be the Keto-Amino one. As can be seen in Fig. 3, this in indeed the case.

Rumer's classes are determined as follows: if the middle base is of the Amino type the class is immediately determined, i.e. a C defines class 4 codons while an A defines class-3, -2, and -1 codons. If instead the middle letter is of the Keto type, we need to observe the character of the first base. In this case, the chemical dichotomy is the Weak or Strong type. A Strong base in the first position defines class 4 codons while a Weak base defines class-3, -2, and -1 codons.

Thus, subsuming the former results, we have a natural means for the classification of the bases inside a codon following the three possible chemical dichotomies and depending on the position of the base in the codon: the third base following the purine-pyrimidine dichotomy, the second one the Amino-Keto, and the first one the Strong-Weak chemical dichotomy. This classification of bases allows for determination of Rumer's classes of degeneracy and the new parity class introduced by the numerical model of the genetic code. Rumer's transformation, i.e. U,C,A,G oG,A,C,U, maps any codon of the degeneracy-4 class to a codon of the degeneracy-3, -2, -1 class, thus, it is a degeneracy-breaking transformation (it reveals an anti-symmetric property of the degeneracy distribution).

Rumer's transformation is a global transformation acting on all the bases of the codon; however, the same effect is obtained if we restrict the transformation to the two first bases of the codon (only the two first bases (quartets) are determinant for the degeneracy class, the third one being irrelevant). Rumer's transformation can be viewed as a composition between the other two possible global transformations (excluding the identity and the transformations maintaining invariant two of the four bases).

These transformations are: U,C,A,G oA,G,U,C, and U,C,A,G oC,U,G,A. The first transformation exchanges bases inside the Strong and Weak dichotomy, and the second, inside the pyrimidine and purine one. Observe that Rumer's transformation corresponds to the exchange inside the third possible chemical dichotomy, the Keto and Amino one. It is interesting to note that the S-W or the Y-R transformation alone changes only one half of the codons of a given Rumer class. Moreover, following the parity rules presented above, it can be seen that the Y-R transformation changes the parity of a codon. This is completely analogous to Rumer's transformation: this transformation can be applied only to the two last letters of the codon with the same effect (the first letter is irrelevant for parity). This property has two interesting consequences; first, it can be hypothesized that a third dichotomy class, being determined by the last letter of a codon and the first of the next one, can be defined. Following the properties for the other two dichotomy classes, Rumer and parity, this class needs to be anti-symmetric under the action of the third chemical dichotomy transformation U,C,A,GoA,G,U,C, (S-W). We have called this class the hidden class and results about it showing a possible connection with a graph theory of anti-codon classes will be published elsewhere. Moreover, the three anti-symmetric transformations together with the identity have a group structure, indeed, they form a 4-Klein group called Klein V group (Jimenez-Montano, 1999; Negadi, 2004; Gusev and Schulze-Makuch, 2004). Second, denoted by pp the probability of occurrence of an odd codon in a real coding sequence, the probability after application of the Y-R transformation will be 1 - pp. On the other hand, if pR is the probability of Rumer degeneracy-4 class, because that the same transformation affects only one half of the codons, if parity and Rumer's classes are independent, the expected probability after application of the Y-R transformation will be one half. The departure from this value is a new indicator reflecting the independence of the two dichotomy classes. Moreover, the same applies for the other two transformations and the hidden class, opening the door to studying also correlations between dichotomy classes of different codons in a natural manner (Gonzalez et al., 2006).

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