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 x_{1}  x_{6}  the variables
accepting values 0 and 1: Let’s group the variables in pairs and designate
by the letters X,
Y, Z:
The matrix encoded this way is written as a triplet
XYZ. 
Each pair of
variables x_{i }x_{j}_{ }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 x_{i}x_{j}, 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 alphahelical regions possessing a strong connectivity (variables
111111), therefore 11 – A;  GUU triplet encodes the amino acid valine
(Val), typical for
betastructural 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. Redbrown blocks (C, U) contain triplets coding for the matrices in which x_{3}=0,
which corresponds to the open acyclic conformations of 4arc graph. In the
dark blue blocks (G, A)
there are the triplets encoding matrices with x_{3}=1,
which corresponds to the cyclic conformations
of 4arc 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:
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 С_{2}): 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 В^{6}, 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 В^{6} (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 В^{6}. ^{ } 
Thus we have shown, that in a basis of the triplet code may lie a coded in the form of
triplets by fourletter alphabet matrix
description of the conformations of 4arc graph
or of the protein pentafragment.
By means of matrixes and
triplets it is possible to code structure of narc 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 4arc graph can be recreated in the structure of the protein. Section
4 of our page is devoted to
it.