## Particular Non Power Number Representation System as a Structural Isomorphism with the Genetic Code Mapping

The palindromic ordering given in Table 6 shows that the degeneracy distribution of the genetic code is compatible with the degeneracy distribution of a non-power representation system. There are not many theoretical results about non-power representation systems (see, e.g. Wolfram, 2002) and, to the author's knowledge, it has not been demonstrated that any palindromic degeneracy list is amenable to such a kind of system. In more mathematical terms we can say that palindromy of the degeneracy distribution is a necessary condition, but it is not known if it is also sufficient.

For the case of the genetic code we have proceeded by trial and error to find a unique solution for the set of positional weighting values of a length 6 binary non-power representation system satisfying the distribution of Table 2, which implies also necessarily matching the degeneracies shown in Table 6. The values of the non-power positional weights are: [1, 1, 2, 4, 7, 8], and this solution, shown in Table 7, is unique up to trivial equivalence classes (Gonzalez and Zanna, 2003; Gonzalez, 2004). It may be remarked that there exists a multiplicative equivalence: all the positional weights can be multiplied by an integer (a scale factor) without affecting the degeneracy distribution (but breaking the contiguity of represented numbers). Allowing the representation of negative numbers, it is also possible to change the sign of one or more positional weights while conserving the degeneracy distribution, and maintaining contiguity of the represented numbers. Finally, it is interesting to note that there does not exist additive invariance, that is, the addition of the same integer to all the weights changes the degeneracy distribution.

The degeneracy distribution corresponding to this unique solution is shown in Tables 7, 8, and 9. It is easy to see on inspection that Tables 8 and 9 are identical, respectively, to Tables 2 and 6. This result is remarkable because the exact degeneracy distribution of the genetic code has been obtained with a one-step mathematical procedure: the nonpower representation of the whole numbers from 0 to 23 by means of binary strings of length 6 (a total of 64) and using a unique set of six positional weights [1, 1, 2, 4, 7, 8].

However, this result does not represent per se a model of the genetic code.

The genetic code maps the 64 codons into 24 elements (amino acids inside quartets plus the stop signal) while the non-power representation maps the 64 length-6 binary strings into the first 24 whole numbers: the two mappings have the same degeneracy distribution.

Thus, from a mathematical point of view, the genetic code and the non-power representation are connected by a structural isomorphism (they are two mappings sharing the same logical structure). But it is known from modelling theory that a structural isomorphism does not necessarily represents a model; it can be a coincidence, albeit a highly fortunate one (the probability of a random origin of a much less restricting property of the code, known as the Rumer's transformation, which is treated in Section 6, has been estimated as being very low indeed, P = 3.09-32 (Zhaxybayeva, 1996)).

Table 7 Non-power representation of the first 23 whole numbers by length 6 binary strings and positional weights 1, 1, 2, 4, 7, 8 The number 7, for example, can be represented by three different binary strings, i.e. (010000; 001101; 001110); also the parity of strings has been shown: even strings are given in grey

Table 7 Non-power representation of the first 23 whole numbers by length 6 binary strings and positional weights 1, 1, 2, 4, 7, 8 The number 7, for example, can be represented by three different binary strings, i.e. (010000; 001101; 001110); also the parity of strings has been shown: even strings are given in grey

 Represented number Length 6 binary strings 8 7 4 2 1 1 8 7 4 2 1 1 8 7 4 2 1 1 8 7 4 2 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 2 0 0 0 0 1 1 0 0 0 1 0 0 3 0 0 0 1 0 1 0 0 0 1 1 0 4 0 0 1 0 0 0 0 0 0 1 1 1 5 0 0 1 0 0 1 0 0 1 0 1 0 6 0 0 1 1 0 0 0 0 1 0 1 1 7 0 1 0 0 0 0 0 0 1 1 0 1 0 0 1 1 1 0 8 0 0 1 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 1 0 0 0 0 0 9 1 0 0 0 0 1 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0 1 1 10 1 0 0 1 0 0 1 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 0 11 0 1 1 0 0 0 0 1 0 1 1 1 1 0 0 1 0 1 1 0 0 1 1 0 12 1 0 1 0 0 0 1 0 0 1 1 1 0 1 1 0 0 1 0 1 1 0 1 0 13 0 1 1 0 1 1 1 0 1 0 0 1 1 0 1 0 1 0 0 1 1 1 0 0 14 0 1 1 1 1 0 0 1 1 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 15 1 1 0 0 0 0 1 0 1 1 0 1 1 0 1 1 1 0 0 1 1 1 1 1 16 1 1 0 0 0 1 1 1 0 0 1 0 1 0 1 1 1 1 17 1 1 0 0 1 1 1 1 0 1 0 0 18 1 1 0 1 0 1 1 1 0 1 1 0 19 1 1 1 0 1 1 1 1 0 1 1 1 20 1 1 1 0 0 1 1 1 1 0 1 0 21 1 1 1 1 0 0 1 1 1 0 1 1 22 1 1 1 1 0 1 1 1 1 1 1 0 23 1 1 1 1 1 1
 Numbers sharing the Degeneracy,i.e. no. of binary strings same degeneracy coding the same number 2 1 12 2 2 3 8 4

Table 9 Explicit correspondence between the 64 length-6 binary strings and the first 24 whole numbers for the non-power number representation with positional bases [1, 1, 2, 4, 7, 8] (compare with Table 6) Degeneracy no.

(length-6 binary strings) Coded whole numbers

 1 0 2 1 2 2 2 3 2 4 2 5 2 6 3 7 4 8 4 9 4 10 4 11 4 12 4 13 4 14 4 15 3 16 2 17 2 18 2 19 2 20 2 21 2 22 1 23

In order to construct a model it is necessary to establish links between the two pairs of sets defining the applications: inside the domains (codons and length-6 binary strings), and inside the codomains (the amino acids and the whole numbers of the representation). Section 5 is devoted to demonstrating that such links can be established, and thus, that a true mathematical model of the genetic code can be constructed.