3.2. Supermatrix of conformations of 4-arc graph and its spatial representation

3.2.1. Construction and properties of the Supermatrix consisting of triangular matrixes

To begin, we will construct the Supermatrix of triangular matrices describing the conformation of the 4-arc graph and study its properties. There are 26 upper triangular matrices, consisting of six variables, i.e., only 64. We have arranged these matrices in increase of connectivity degree of the graph described by them. The arrangement of matrices can be carried out in different ways, for example, so that the adjacent matrix in rows and columns differ from each other by one bit [5]. However such principle of an arrangement comes to the contradiction with principles of assignment triplets and amino acids (see the section. 3.3.), therefore the preferred variant was an arrangement of matrices in the form of blocks (fig. 24). Let's consider the basic properties of the block Supermatrix.

 

BLOCK STRUCTURE

The principle of ordering of triangular matrices in the form of blocks is based on the fact that matrices of each block are grouped on the basis of a community of the second pair variables - x3x4. (see Section 3.1.2.).  In this case, as shown in Figure 24, in the first block (at the left above), in the upper left corner is the matrix A, describing non spiral conformation (fully disconnected graph) that contains only zeros, and the fourth block (on the right below) in the lower right corner settled matrix A ', describing the helical conformation of the graph (fully connected graph) consisting of all ones.

 

 

 

 

 

 

 

The Supermatrix consists of 4 blocks (Fig. 24).

 

In each block consisting of 16 matrices, the second pair of variables x3x4 is identical. In accordance with the values ​​of these variables blocks are designated by large figures:

00, 01 (the upper blocks)

10, 11 (the bottom blocks).

 

In blocks 00 and 01  x3 = 0. These blocks are marked with a red-brown color. In blocks 10 and 11   x3 = 1. They are painted dark blue.

 

In each block in the rows are the matrices having the same first pair of variables, they are shown on the left side. The order of raws 00, 10, 01, 11.

 

The columns are matrices with identical third pairs of variables, which are shown above. The sequence of columns 00, 01, 10, 11.

 

 

Fig. 24. Construction of a block supermatrix from 64 triangular matrices describing the conformation of the 4-arc graph.

SYMMETRY IN BLOCKS

 

 

 

 

 

 

 

 

 

On the main diagonal of each block (for example, the block 00) are matrices in which the first pair of variables is symmetric to the third pair: 000000, 100001  etc.

Matrices in which the first pair of variables is symmetric to the third pair are located symmetrically concerning the main diagonal  for example 100000 and 000001, 010000 and 000010, etc.

 

Fig. 25. Symmetry of triangular matrixes in the block 00.

ANTISYMMETRY

 

 

 

 

Within the supermatrix each matrix has the antipode in which values 0 turn to 1, and 1 - to 0, i.e. 0 <---> 1. The matrices connected by these transformation are located rotational symmetric (group of symmetry С2), for example:

000000 and 111111 (A и A'),

000001 and 111110 (B и B'),

000010 and 111101 (C и C').

Two groups of antisymmetric matrices can be identified in different ways. We have chosen variant (the Russian letter "Г", red-brown and dark blue, nested) is justified below.

 

 

 

Fig. 26. Antisymmetry of triangular matrices in the Supermatrix.

The variant of Supermatrix cut chosen by us at which matrices of the top part will be transformed to matrices of the bottom half by replacement  0 <---> 1, coincides with the natural division of the hypercube B6 into two halves ("red" and "dark blue"), which are combined when rotated on 180о.

In this Supermatrix, each matrix contains 6 variables and may be connected with six other matrices differing from the original one by a single value (one-bit transitions). This is consistent with the principle of constructing a Boolean hypercube B6, shown below.

 

3.2.2. Spatial representation of a Supermatrix: Boolean hypercube B6.

 

ARRANGEMENT OF QUARTETS OF MATRICES

 

The series of matrices from the blocks located in a "red-brown" part of the Supermatrix, are in the Hypercube (Fig. 27) in "red" quartets М11 and М21.

The series, consisting of blocks of "dark blue" part of the Supermatrix are in "blue" Hypercube quartets М12 and М22.

Quartets of the matrices possessing symmetry of the first and the third pairs of variables are marked by yellow color on a hypercube.

 

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Fig. 27. An arrangement of quartets of triangular matrices on the Boolean hypercube B6.

ANTISYMMETRY TRANSFORMATION

 

 

 

 

 

 

 

 

 

 

 

 

Triangular matrices connected by transformation of antisymmetry (0<--->1), are arranged symmetrical (group of symmetry С2) in the "red" and "blue" halves of the Hypercube (Fig. 28).

 

 

For interested persons who want to reflect there is a complete structure of Нypercube В6 on a white background which can be printed.

 

 

Fig. 28. An arrangement of matrices connected by transformation of antisymmetry on the Boolean hypercube B6.

 

 

 

3.2.3. Properties of 4-arc graphs in blocks of the Supermatrix

 

Graphs can be characterized by the number of degrees of freedom, i.e. possibility of graph elements moving in three dimensions. Since the graph is connected in a chain, then its elements can have two degrees of freedom as maximum, such as moving from side to side or front to back. One degree of freedom and zero degree are possible also. In the latter case, all elements of the graph, for example, vertices, fully fixed. We consider the properties of graphs in each of the four blocks of the Supermatrix. A protein chains can be described similarly.

 

In blocks 00 and 01 x3 = 0. It means, that between vertices i and i-4 there is no edge of connectivity and the graphs of these blocks have not closed conformations.

BLOCK 00

 

 

On the main diagonal of each block, for example 00 (Fig. 29), the graphs having internal symmetry are situated:

000000, 100001, 010010, 110011.

 

On both sides of the main diagonal the graphs symmetric to each other are located, for example:

100000 and 000001, 000010 and 010000 etc.

 

In the block 00 are mostly weakly connected conformation, with the number of degrees of freedom 2 or 1. The only graph of the block, with zero degree of freedom is described by the matrix 110011.

 

Fig. 29. Weakly connected conformations of 4-arc graphs in block 00.

BLOCK 01

 

 

 

 

 

 

 

 

In this block are mainly located the folded conformations, with number of degrees of freedom 1.  A typical example of such a conformation is a graph described by the matrix 100101, which corresponds to the beta-structure conformation of the protein (see Section 3.1.3.).

 

The only graph of the block, with zero degree of freedom described by the matrix 110111. It corresponds to the helix 310, discussed in Section 3.1.3.

 

 

 

 

 

Fig. 30. The folded conformation of the 4-arc graphs in block 01.

In contrast to earlier, in blocks 10 and, discussed below, x3 = 1. Between vertices i and i-4 of these graphs there is an edge of connectivity and therefore they are cyclic.

BLOCK 10

 

 

Block 10 (Fig. 31) contains a weakly connected cyclic conformation of the graph with the degree of freedom 1 and 2.

 

Three graphs have a fixed conformation (degree of freedom 0) 011011, 111010 and 111011. By their structure, they are close to the conformation of the protein alpha-helix (see in Section 3.1.3.).

 

Fig. 31. Movable cyclic conformation of 4-arc graphs in block 10.

BLOCK 11

 

 

 

 

 

 

 

 

 

 

 

 

 

 

In block 11 also are cyclic conformation of the graph, but they have a much lower degree of freedom.

 

Eight of them - 101101, 101110, 011101, 011111, 111101, 111110 and 111111 are fixed (the degree of freedom is equal to zero) and correspond to the conformation of the protein alpha-helical.

 

 

 

Fig. 31. Cyclic conformations of 4-arc graphs in block 11.

 

These graphs and their matrix description, however, are not suitable for storage, transfer and reproduction of structural information contained in them. For these purposes, is much more convenient linear arrangement of elements, which can be achieved after transformation of Supermatrix in triplet code (Section 3.3.).

 

 

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