2.2. Spatial structure of the doublet genetic code

For the first time the spatial structure of the doublet genetic code was proposed in [14]. It is impossible to exclude, that authors were inspired for construction of this structure by our rhombic variant of the genetic dictionary [12] quoted by authors and, in a modified form, even reproduced by them. An alternative doublet code structure was described also in [3]. The disadvantage of these variants was absence of visual representation of the symmetry of the two groups of doublets connected by Rumer’s transformation  [13]. Model of spatial structure of doublet code proposed in our paper [4] eliminates this defect.

 

2.2.1. Principles of model construction

 

It is possible to construct two types of connectivity graphs in the form of tetrahedrons of the four nitrogenous bases, in which a sequence of C, G, U, A, is anti-clockwise (L tetrahedron) and clockwise (D-tetrahedron):

 

It is possible to allocate three types of single transitions between the bases:

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

between the complementary bases
(solid lines)

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

between non-complementary bases
(a rare dotted line)

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

transitions of pyrimidine - pyrimidine and purine – purine (frequent dotted line)

"Rhombus" doublets also can be connected with each other by single transitions, and each neighboring doublet differs from an initial doublet on one basis, for example CC <--> GC:

 

 

 

At the left there are doublets with substitutions in

the first position

 

 

 

On the right there are doublets with substitutions in  the second position

 

Continuing and further deriving of doublets on the basis of single substitutions (single transitions), we finally build the following structure (Fig. 3):

 

 

 

 

 

 

 

We call it the L-structure.

There is also a symmetric D-structure.

 

The resulting L-structure, as mathematicians say, is isomorphic to the four-dimensional Boolean hypercube (designation B4) superimposed with additional lines along the diagonals of the parallelograms.

 

Such structure is called a "six-dimensional simplex." Each doublet, located at the vertex of the hypercube is connected by single transitions with six neighboring:

 

 

 

Example:

 

6 lines proceed from doublet СС connecting 6 doublets. These lines are allocated by red color.

 

 

Fig. 3. The spatial structure of the doublet genetic code, isomorphic to the Boolean hypercube B4.

 

 

 

2.2.2. Properties of the doublet genetic code revealed by its spatial structure

 

A. Single transitions

Transitions C<-->G, U<-->A are united into parallelograms lying in a horizontal plane (Fig. 4)

Transitions C<-->U, G<-->A form parallelograms located in a vertical plane.

Transitions C<-->A, U<-->G connect diagonals of parallelograms of the previous two types

 

 

 

 

Fig. 4. Single transitions C<-->G, U<-->A on structure of the doublet code.

Fig. 5. Single transitions C<-->U, G<-->A in the spatial structure of the doublet code.

Fig. 6. Single transitions C<-->A, U<-->G in the spatial structure of the doublet code.

 

B. Rumer’s transformation

C. The compact arrangement of amino acids. Cycles

The doublets, coding for one and two amino acids and related  by Rumer’s  rule (C<-->A, G<-->U), occupy a symmetrical position in the hypercube, for example:

AC<-->СA, GC<-->UA, GG<-->UU, GU<-->UG, etc.

The amino acids similar in structure are located in a hypercube compactly and enter into cycles: Pro, Ala, Gly, Arg; Ser, Thr, Ser-Arg, Cys-Trp; Leu, Val, Ile-Met, Phe-Leu;  His-Gln, Asp-Glu, Asn-Lys, Tyr.

 

 

 

Fig. 7. The position of doublets connected by transformation of Rumer, in a doublet structure of the genetic code.

Fig. 8. The location of amino acids in the spatial structure of the doublet genetic code.

 

After becoming acquainted with the principles of construction of the spatial structure of the doublet code it will be easy to understand how to construct the spatial structure of the triplet genetic code (section 2.3.).

 

 

 

 

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