3.3. Supermatrix transformation into the triplet topological code

3.3.1. Coding of triangular matrices

As mentioned at the end of Section 3.2., for the purpose of storage, transfer and reproduction of structural information contained in the matrices, it is more convenient to use a linear arrangement of elements, which can be achieved after Supermatrix transformation into triplet code.

 

 

Let's consider again the matrix of the six elements, where x1 - x6 - the variables accepting values 0 and 1:

Let’s group the variables in pairs and designate by the letters X, Y, Z:


x1x2 = X,  x3x4= Y,  x5x6 = Z.

 

The matrix encoded this way is written as a triplet XYZ.

 

 

Each pair of variables xi xj can accept following values:

 

 00, 01, 10, 11.

 

 

They can be encoded by means of the four letters alphabet, for example, the Latin:

 

 A, B, C, D

 

We use for these purposes the first letters of names of the nitrogenous bases of the genetic code:

C, U, G, A

The question of that the letters of the genetic code may correspond to a pair of Boolean variables xixj, was raised long ago [7]. The section in the paper [23] was dedicated to the analysis of various variants of such correspondences.

 

The choice of correspondences of  triplet letters to pairs of Boolean variables

 

At a choice of these correspondences, we started from the idea that the properties of the encoded triplet conformation should coincide with the properties of amino acids that corresponds to the triplet:

-Triplet containing the CСС, which encodes the antispiral amino acid proline (Pro), relating to the sites of the protein with a weak connectivity (matrix 000000),

therefore, should be 00 – C;

- AAA triplet encodes an amino acid lysine (Lys), often occurring in the alpha-helical regions possessing a strong connectivity (variables 111111),

therefore 11 – A;

- GUU triplet encodes the amino acid valine (Val), typical for beta-structural regions of the protein (described by variables 100 101), which can be obtained if 10 - G, and 01 - U.

 

As a result, the following correspondences have been chosen:

00 - C

01 - U

10 - G

11 – A

Based on selected correspondence by substitution to the triangular matrixes the letters of a code instead of pairs of variables the Supermatrix was transformed into a triplet topological code (Fig. 32.) , completely identical to a genetic code.

 

3.3.2. Properties of the table of a triplet code transformed from the Supermatrix

 

BLOCKS OF TRIPLETS

 

 

 

 

As seen in Figure 32, blocks of matrices with a common second pair of variables transformed into blocks of triples with a common  second letter:

 

C <---> 00  U <---> 01, G <---> 10, A <---> 11.

 

Red-brown blocks (C, U) contain triplets coding for the matrices in which x3=0, which corresponds to the open acyclic conformations of 4-arc graph.

In the dark blue blocks (G, A) there are the triplets encoding matrices with x3=1, which corresponds to the cyclic conformations of 4-arc graph.

 

 

 

 

Fig. 32. Supermatrix transformation in a triplet topological code

SYMMETRY IN BLOCKS

 

 

 

 

 

 

On the main diagonal of blocks (as an example on Figure 33 the block C is shown), there are triplets coding for symmetric matrices and graphs.

 

Since C = 00, A = 11, then in the first and the third position these letters encode the symmetric state of connectivity (triplets CCC and  ACA).

 

In turn, G = 10, and U = 01, so the letter G in the first position encodes the state of connectivity symmetric to encoded by the letter U in the third position and vice versa (triplets GCU and UCG).

 

On both sides of the main diagonal are the triplets coding for symmetric matrices and conformation, for example:

 

GCC - CCU, UCC - CCG, ACC - CCA, etc.

 

Fig. 33. The symmetry in the triplet code blocks.

ANTISYMMETRY

 

 

 

 

Аntisymmetry transformation looks as follows:


C <---> A, G <---> U,

 

that corresponds to the Rumer”s transformation in the triplet genetic code (see Section 2.3.). In this case the antisymmetric triplets occupy a symmetrical position of the table (Group of symmetry С2):

CCC <--> AAA, CCU <--> AAG, CCG <--> AAU    etc.

 

 

Fig. 34. Antisymmetry in structure of the triplet code

SPATIAL REPRESENTATION OF THE TRIPLET CODE

 

Spatial representation of the given system of triplets is the structure, isomorphic to the Boolean hypercube В6, identical to spatial structure of the triplet genetic code (Section 2.3.).

It can be obtained on the basis of the transformation of matrices located at the vertices of the Boolean hypercube В6 (see Section 3.2.2.), into the triplets using chosen correspondences of pairs of variables to letters of triplets.

 

Thereby we have answered the question posed in Section 3, why the structure of the triplet genetic code has appeared isomorphic to Boolean hypercube В6.

 

 

Thus we have shown, that in a basis of the triplet code may lie a coded in the form of triplets by four-letter alphabet matrix description of the conformations of 4-arc graph

 or of the protein pentafragment.

 

 

By means of matrixes and triplets it is possible to code structure of n-arc graphs in the form of triplet sequence. However, hopes rested upon this description, that by means of the offered algorithm it will be possible to predict protein structure, were not justified. Therefore in the updated version of our page we do not place this algorithm. Interested persons can read about it in our monograph [8].

 

It is crucial, how coded in the form of triplets conformation of 4-arc graph can be recreated in the structure of the protein. Section 4 of our page is devoted to it.

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