Cracking the genetic code(s) with a modular determinative degree: An algebraic approach

Abstract

ABSTRACT:We introduce a new "modular" (compositional and positional) determinative degree for the four bases U, C, A, and G to define a "Dinucleotide charge number" for the dinucleotides, which discriminates between the two octets of dinucleotides: M 1 and M 2 . These two components are exchanged under a transformation, which, we show, implements the Rumer symmetry. We invoke also the base "size index" of Rosen as a complementary determinative degree for the third-base component of a codon. Next, we define the address functions for the 64 codons (amino acids), which are functions only of the composition and position of these codons in the genetic table. With these ingredients, we build an algebraic classification of the amino acids in the genetic code and some of its nonstandard versions as a preliminary application.

Rumer's transformation, in biology, as the negation, in classical logic

“…In Ref. 3, we started from the constituents of the four nitrogenous bases, that is, the hydrogen, carbon, oxygen, and nitrogen atoms, written as 2 × 2 matrices [see Eq. (7)], and were led, after constructing the molecules U , C , A , and G using ordinary matrix multiplication, to the following base matrix: In this article, we improve the formalism by using a new concatenated “Kronecker”‐like product, defined in the Appendix, to take into account the noncommutativity of the bases (e.g., UA is not the same as AU ).…”

“…It is well known that M 1 contains doublets for which the third base in the triplet (codon) is irrelevant (for coding an amino acid) while in M 2 the knowledge of the third base is necessary (except, of course, for the singlets). Comparison of (3) and (5.1) shows that the transformation R 1 effectively implements the Rumer symmetry ( CUGA ↔ AGUC , see 3). It is transparent from Eq.…”

“…(12). To see the connection with the biologic objects of Section 2, let us, first, recall the four nitrogenous bases in matrix form 3: with The capital letters representing the atoms are themselves given by 2 × 2 matrices (see 3), and the base matrix ℬ︁ [Eq. (1)] is constructed as U + C + + .…”

“…Now, the connection with the bases could be seen as follows. Let us invoke the atomic content of the four bases and their Rosen's “size index” 9, in an extended form, 3, to include thymine [ T : ( C ) 2 C 2 6 O ]: This number captures the monotonic size gradation of the bases and is a consequence of their relative atomic content [see Eq. (7)], where the powers give also the relative number of each atom type.…”

“…At this point it is interesting to note that it is indeed possible to construct a matrix having, as matrix elements, the above numbers 0, 1, 2, and 3. It suffices to add a parameter, a , in the first matrix element in the atomic matrices (see the appendix of 3): ε = a [ Z i + 2( j − 1)]. In this way, a = 1 corresponds to the ordinary case of the atomic matrices with their atomic numbers while the case a = 0 (and i = 1) leads to the following simple matrix, written also in binary notation: This matrix could represent either the symbols in (9) and (9.1), codifying the size indices of the bases, or those in (10), codifying the building blocks of logic.…”