Arithmetic in Gamows Context

Gamow (1954) was the first who predicted the genetic code itself, its 64 base triplets, 20 canonical amino acids, and average degeneracy. Being a theoretician in the physical sciences, he searched for a universal formula of the genetic code. He stopped that search after biochemists had deciphered the code (Nirenberg et al., 1965). In the meantime, the universal formula exists and is described below.

The same numbers - 20 amino acids and 20 combinations of four bases, three at a time - encouraged Gamow to suggest the very first stereochemical model of the code origin. It turned out that his model is incorrect, but a remarkable coincidence remained. Now, Gamow's 20 combinations give rise to the numerous balances and their decimal syntax. The combinations - placed onto triangular substrates in Fig. 7 - become triplet makers now. Each combination makes its triplets - base after base - by circular motion or spin around the substrate in two opposite directions. A pair of circular arrows symbolizes these spins. New Spin o Antispin transformation as well as Rumer's and two 50% transformations apportion the 64 life-size triplets relative to the central axis (see for detail shCherbak, 2003).

According to Gamow's research approach, the combinations were grouped into three sets. This was done according to the composition of identical and unique bases, regardless of their type and position inside a triplet. The joint first and second sets of triplets having three identical and three unique bases show equilibrated nucleon sums 666 + 999 of whole molecules in Fig. 7a. There is also a new chain-to-chain balancing mode in this balance with equilibrated sums 703, i.e. PN 19 x PN 037 (see Section 14).

The third set of triplets with two identical and a unique base are shown in Fig. 7b. There are two subsets bisected by the vertical central axis whose triplets have either two identical pyrimidine or two identical purine bases. These subsets form a balance with 999 nucleons in each equilibrated arms. The equilibrium appears again between the two sets of side chains. No standard blocks take part in the chain-to-chain type of balancing. However, the imaginary borrowing acts precisely even in absence of standard blocks.

Note that the vertical central axis has bisected the subsets in a way that disregards the type of unique base. Restoring symmetry the horizontal central axis bisects the same triplets using the type of unique base and, this time, disregarding the type of identical ones. Now, the side chain nucleon sum of the triplets whose unique base is pyrimidine equals to 888. Therefore, decimalism is free to act without balance too.

Let us return to the 999-and-999 balance. Its right arm is thrice equilibrated. These three regularly organized arms consist of 333 nucleons each. One of the arms - complete line - has a cloned line written by synonymic triplets at the bottom of Fig. 10b. The cloned line of the synonymic triplets code for the same amino acids, hence, for the same 333 nucleons. The cloned line splits the corresponding subset into summands 333 + 777. These summands can be represented as a balance 777 + 777 of the block + chain-to-chain type: add the 333 side chain nucleons to its own 6 x 74 = 444 nucleons in the standard blocks.

Besides Gamow's "context" there are a few standard representations among complementary life-size triplets categorized by total quantities of their hydrogen bonds. These quantities can be equal to integers 9, 8, 7, or 6. There are 24 triplets having seven hydrogen bonds in the genetic code. Their total sum of side chain nucleons can split regularly into balanced parts 333 + 333 + 333 + 333 or 444 + 444 + 444.

Metaphorically, in our denary dialect of the arithmetical language we have written down the names of certain cardinal numerals of the genetic code. This is the answer the genetic code gave: These inscriptions are their inborn names. The names have been given them in the time of genesis. Now, in their native numerical language these names are reproduced in written form again because the current dialect turns out to be identical to that primordial numerical language.

Fig. 7 Gamow's division of the genetic code. Pythagoras' images symbolize balanced summations and their decimal syntax. The proline imaginary borrowing turns balanced sums into virtual values.

Gamow's 20 combinations are placed on triangular substrates. Each of the combinations makes certain triplet(s). The combinations are divided into three sets depending on their base composition. Four combinations with identical bases make four triplets for the first set. The other four combinations with unique bases make two dozens of the triplets for the second set. Twelve remaining combinations make 36 triplets with two identical bases and a unique one and form the third set (shCherbak, 1996). (a) There is a balance that comprises the first and second sets. The Spin ^ Antispin transformation does not affect the triplets of the first set, but it apportions the triplets of the second set. The balance appears when Rumer's transformation is paired with 50% transformation T ^ C, A ^ G in the first set and with 50% transformation T ^ A, C ^ G in the second set. Both sets are coaxially placed in one of the alternative positions. The balance is true for the universal genetic code

Fig. 7 Gamow's division of the genetic code. Pythagoras' images symbolize balanced summations and their decimal syntax. The proline imaginary borrowing turns balanced sums into virtual values.

Gamow's 20 combinations are placed on triangular substrates. Each of the combinations makes certain triplet(s). The combinations are divided into three sets depending on their base composition. Four combinations with identical bases make four triplets for the first set. The other four combinations with unique bases make two dozens of the triplets for the second set. Twelve remaining combinations make 36 triplets with two identical bases and a unique one and form the third set (shCherbak, 1996). (a) There is a balance that comprises the first and second sets. The Spin ^ Antispin transformation does not affect the triplets of the first set, but it apportions the triplets of the second set. The balance appears when Rumer's transformation is paired with 50% transformation T ^ C, A ^ G in the first set and with 50% transformation T ^ A, C ^ G in the second set. Both sets are coaxially placed in one of the alternative positions. The balance is true for the universal genetic code

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