4. Antisymmetry in the system of canonical amino acids


4.1. Antisymmetric properties of the side chains of the canonical set of amino acids


Pay attention, that among the amino acids side chains we find pairs.


1.There are a pairs of amino acids with similar properties but different chain lengths:

- aspartic (Asp) and glutamic (Glu) acids (Table)
- valine and isoleucine,
- serine and threonine,
- asparagine and glutamine, arginine and lysine.



2.Some pairs of amino acids differ in their properties, but have similar size:

- serine (Ser) and cysteine ​​(Cys) (Table)
- aspartic acid and asparagine,
- glutamic acid and glutamine
- histidine and phenylalanine,

- tryptophan and tyrosine.





Pairs (1) have a quasi-mirror antisymmetry.

Pairs (2) possess not mirror antisymmetry.

3. Some amino acids form a pair on the basis of opposites of their properties:

aspartic acid (Asp) has acidic properties, and arginine (Arg) - alkaline (table), similarly, a pair of glutamic
acid and arginine.





4. It is possible to oppose a number of amino acids on weight:

- short amino acid serine (Ser) can be contrasted with the massive amino acid - tryptophan (Trp) (Table),

- more heavier - threonine (Thr) – to lighter than the tryptophan - histidine (His).





Pairs (3) are antisymmetric in charge.

Pairs (4) are antisymmetric by weight.


All of these types of antisymmetry will be visually presented as a result of construction of our system of amino acids.


4.2. Antisymmetry, revealed by system of canonical amino acids


In the course of construction of system of amino acids on the dodecahedron we explicitly or implicitly used the principles of antisymmetry. Now, when the system is constructed, we will consider the general features of antisymmetric groups of amino acids. For this purpose, for each type of symmetry, we will use different color backgrounds in the circles, in which the side chains of amino acids are situated.


4.2.1. Quasi-mirror antisymmetry


Antisymmetric amino acids, which are located on both sides of the plane I, we call quasi-mirror plane. They form pairs, which have similar physical characteristics but different chain length or weight.


They are connected by the transformation, which we designate by the letter alpha (a).


To the right of a plane I there were shorter or lighter side chains (they are shown on light green background), and at the left - longer or more heavy (dark-green background). To see this in more detail, please see 4.2.1.

Fig. 19. Coloring of amino acids connected by the quasi-mirror antisymmetry.
To the left of the plane I are circles of light green color - the lighter side chains, to the right of the plane I – circles of dark green color - the more massive ones


4.2.2. Not mirror antisymmetry


The side chains of amino acids having not mirror antisymmetry, are located on the dodecahedron on opposite sides of the plane II: behind the plane II (background in the circles is painted in pale pink), and prior to that plane (dark pink background).


For brevity we shall speak:  behind a plane and prior a plane, the name referring to the plane II.


They are connected among themselves by transition which we have designated by the letter beta (b). In this case the side chains in pairs usually have similar size but different physical properties. In the section 4.2.2. pairs of side chains of amino acids discussed in more detail.

Fig. 20. An arrangement of the side chains in pairs of the amino acids connected by not mirror antisymmetry.

Behind a plane II - circles of pale pink colour, before a plane II - dark pink colour.


4.2.3. Rotary antisymmetry


This type of antisymmety arises for two groups of side chains, which are located above and below the plane III, separating the dodecahedron into two parts: upper and lower.


If the upper part of the dodecahedron, in which the side chains of amino acids are represented in the circles of pink, rotate around an axis in the plane III, the vertices of the upper part coincide with the vertices of the bottom part of the dodecahedron, in which the side chains of amino acids placed in the light blue circles.


The resulting pair of side chains are connected among themselves by transformation of rotation, which we denoted by the letter gamma (g). They have opposite properties and the different size. In section 4.2.3. these pairs are analyzed in more detail.

Fig. 21. Position on the dodecahedron of side chains of amino acids connected by rotary antisymmetry.
Above, over the plane III - pink circles, below, under the plane III - light blue colour.


4.2.4. Antisymmetry of complementarity


If we connect the opposite vertices of the dodecahedron by diameters, the side chains of amino acids located at these vertices form a complementary pairs.


This type of antisymmetry may be named as antisymmetry of complementarity.


The resulting pairs of side-chains linked by three transformations:  a,  b, and g.


They have as much as possible opposite size and properties, and if we compare the pairs, starting from the right side and left side, then these properties as would be compensated. More detailed analysis of the antisymmetry of complementarity is done in Section 4.2.4.

Fig. 22. Position on the dodecahedron of side chains of amino acids connected by antisymmetry of complementarity.
Opposite vertices of the dodecahedron are linked by diameters.
Circles in the opposite vertices of the dodecahedron have different coloring, symbolising the complementary of side chains of amino acids located in them.


4.3. Transformations of amino acids at transition through antisymmetry planes


Since we have three planes of antisymmetry transition to antisymmetric amino acid through these planes can be designated in the same way as in section 2.3., by any letter. Following designations have been entered:

- amino acid transition in itself is designated by figure 1.

- transition through a plane I - the letter a (alpha),

- transition through a plane II - the letter b (beta);

- rotation around an axis laying in a plane III - the letter g (gamma).

Then it is possible to describe all transformations of the amino acids located in a column 1, in each group in the form of the following table.











Subgroup 1









Subgroup 2









Subgroup 3









Subgroup 4










So, we have shown that a system of canonical side chains of amino acids, constructed by us, allows to visualize the four types of anti-symmetry, which exist for them: quasi-mirror, not mirror, rotary and antisymmetry of complementarity.


From this fact we can make the following conclusion:

possibility of imposing (superposition) of all four types of antisymmetry for the given system of amino acids testifies that the offered variant of spatial structure of amino acids on the dodecahedron is unique. It is impossible to carry out a permutation of amino acids without affecting any type of antisymmetry


Do not believe me - try it yourself.


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