The amount of correlations between the genetic code and numeral series is difficult to regard as only random ones.
   A general problem is of course that it still doesn't seem to exist any known biochemically accepted mechanisms that could "explain" construction along such numeral series, however established facts in the other mentioned examples. It could however be questioned in which sense the 2x2-series behind the periodic system is "explained", or the formula for spectral lines of hydrogen.) Facts are there. Science has only its models, as far as possible congruent with the facts.

With the hypothesis here that they really reveal features in how Nature organized the genetic code, what should it imply? About the elementary series 5 →> 0, the series of valences for atoms in the genetic code could be remembered: P - C - N - O,S - H = valences 5 - 4 - 3 - 2 - 1. A dimensional interpretation seems inevitable, with regard to exponents and to transformations between nb-systems.

How should the exponent 2/3 be explained? We have squares in the 2x2-chain behind the periodic system and intervals between inverted squares behind the spectral lines of hydrogen. These formulas concern electron shells of atoms, i. e. the property charge. With mass and charge most elementary assumed as a mutual relation D3 to D2, cubes become natural. We have mass as the energy form concentrated in atomic nuclei, charge expressed in the atomic shell with released energy in kinetic form. Why then inverted cubes? They lead inwards to a deeper level, as does the inward direction toward nucleus in an atom.
    One association goes to such celestial bodies as white dwarfs, where mass is inversely proportional to the cube of the radius. (Another may be the general feature of atoms in a full shell of the periodic system, in which the radius decreases towards heavier nuclei.)

The many relations of disparate kinds to the 2x2-chain and other simpler chains support the interpretation of the genetic code as built on an elementary chain x = 5 - 0 with exponents of different degrees. With a dimensional view on the exponents, it could imply, either that such chains preceded the more elaborated ES-chain when the coding system emerged or could be regarded as simultaneously existing on underlying levels. It's possible to imagine a dimensional development from both ends of the chain towards step 3 - 2 in the middle with increasing agreement of mass distribution in the genetic code:

x4 x3 [ x3/2 →← x2/3] x2 x1.

The mass distribution as described in section I often implied minus/plus lower numbers in the ES-series, correlating with features in the background model. It points to a two-way direction in he chain of both disintegration and synthesis. This could seem to conflict with the common view on evolution as a stepwise synthesis towards more complex and bigger units. Yet, a double-direction is natural in Nature, if we think of macrocosm, Big Bang and both processes in celestial Hx-clouds. It could be mentioned that even among physicists this opposite view of disintegration, starting from a whole, has been proposed. (There is a similar pattern of two-way direction in the protein synthesis, where tRNAs as from opposite strands of DNA meet mRNA "the other way around" at ribosomes in the "middle" of the process.) See figure here, with dimensional interpretation of the forms from double direction (D4) in DNA to single-strnded RNA as vector (pole 4b) outwards to ribosomes (D3) - meeting tRNAs (as "clover leaves" D2) and ams.

It's shown too that not only mass distribution on codon groups of ams correlates with the ES-chain but also other bases for mass division, for instance with main groups of atom kinds and the not codon-dependant B-chains as well as with several features in the origin of ams from stations in glycolysis - citrate cycle. This suggests an interpretation where the same principle scheme is developed on different levels or as representing different axes in a coordinate system when the genetic code emerged.
   The single fact that the mass division on C-skeleton and other atoms (960 and 544) is the same as between main codon groups (U+A, 960 and G+C, 544) supports in itself the general suggestion that the code is built on a numeral series.

In several ways the results seems to agree with the coevolution theory [6, 7]. There is the relation with biochemical origins of ams from glycolysis and citrate cycle. There is the view of codon domains as totals, differentiated in following steps, even if the "codon domains" here is related to mass sums of ams. There is also the fact that G1-coded ams "arrive first" in the number chain as 5 out of about 7 ams assumed first in that theory: GG-GC-GU-GA-GA besides Ser UC and Phe UU..

Then about mass again
, rejected as irrelevant for codon assignments: In addition to arguments in the Introduction it's reasonable to ask for instance why precisely these ams have been selected for coding, not other ones? The selection seems rather random. Why just this number of ams with oxygen as end groups, that number of ams with nitrogen? (Besides that both types and polar and non-polar ams surely have been necessary.)
    Further, when much research in this field has been focusing on the "most stable" configuration of the coding system, one could naturally ask what the background is for this stability? One aspect is of course that the most common isotopes have shown up to be most stable. (When calculating with common mix of isotopes today, atomic weights should change the sum of R- plus B-chains of ams from 3276 →> 3280 abbreviated, R-chains from 1504 →> 1506, no more than the deviations of single units (u) in this analysis.) In addition, the analysis here mostly concerns groups of ams, i. e. sums were an individual deviation in mass might have a rather small influence.
   The fact that Ileu sometimes gets mixed with Leu by tRNAs could also be mentioned, differing in structure but having the same mass and atoms.

Does the proposal for a guiding numeral series exclude such an individual invention among certain organisms as Pyl, called the 22nd ams, occupying a stop codon? Pyl adds 108 to R-chain of Lys, i. e. the interval 3' to 1' in the ES-chain and could eventually be suspected as a "misreading" of the chain, leading to a compound, a new "word"?

The examples of transformations between nb-systems are astonishing and certainly provocative. They support however a general dimensional view in the creation of the code and actually too the relevance of the ES-chain. They seem to reveal a deep level in the reference system of a hitherto unknown kind, representing the very steps between dimensional degrees. In physical and biochemical terms they should imply something like mutual resonances between "mass fields" in different dimensional degrees, relations and fragmentation guided by geometrical and arithmetical rules. A problem is naturally the superfluity of such possible transformational relations.

If proposals in this paper are accepted as hypotheses, they will naturally raise many new questions and lead to secondary hypotheses, which in their turn could be possible to test. The dimensional aspects, mostly omitted here, should reasonably, if elaborated further, have implications for protein structures and their different functions in cells.
   Whatever to believe about the arithmetic, something of that kind resembles life
- in being very simple and very productive - and naturally multidimensional.







© Åsa Wohlin
Individual research
E-mail: a.wohlin@u5d.net


Links and Notes

Table 24 amino acids (ams)
R-chsins, A, Z, N

- ways of writing -

Background model

Files here:

0. Amino acids and codon bases.
Why this coding system?

All these files in 3 documents, pdf:
Section I, files 0-11

Section II, files 12-16
Section III, files 17-22
Discusssion, References in section III

To 17 short files.
- partly other material -

The 17 files as one document,


An earlier version (2007)
with more material
on the same subject, 73 pages

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