Subgroup 1

Subgroup 2

Elements A and –A are connected by transformation g and are in a plane I at the top and bottom (fig. 4).

The canonical set has only two simple amino acids - the antipode: glycine (Gly), has no side chain, and proline (Pro), which is built into the backbone of the protein.

They are placed respectively in the top and bottom vertices of a dodecahedron instead of elements A and –A (fig. 8).

The subgroup 2 contains elements B and –B. On the dodecahedron element B is connected by an edge with element A (fig. 4). Amino acid alanine (Ala), with the shortest non-polar side chain is the most close  to glycine. Leucine (Leu), which has a long nonpolar chain, may be its antipode.

Position of amino acids of a subgroup 2 on the dodecahedron is shown in Figure 9.

Fig. 8. An arrangement of amino acids of a subgroup 1 on the dodecahedron.

Fig. 9. Arrangement on the dodecahedron of amino acids of a subgroup 2

Four amino acids of a subgroup 3 occupy four vertices at the top of a dodecahedron, and others four - in the bottom.

At the top vertices may be the nearest on the size to glycine side chains. Two of the shortest amino acids serine and threonine were located instead of the elements ¹A and A₁.

Both amino acids contain hydroxyl groups (red atoms). Serine (Ser) was located on the dodecahedron to the right of the plane I, and threonine (Thr) - to the left (fig. 10).

More massive side-chains - methionine and cysteine have been placed on the right and on the left instead of elements ²A and A₂ (fig. 11).

These amino acids contain atoms of sulphur (yellow). Cysteine (Cys) was placed on the dodecahedron to the right of the plane I, and methionine (Met) - to the left.

Fig. 10. An arrangement of the side chains of serine and threonine in the model on the dodecahedron

Fig. 11. Position on the dodecahedron of side chains of cysteine ​​and methionine.

Cyclic amino acids that have the most massive side chains, were placed at the bottom.

Near to the proline, having five-membered cycle, instead of the elements –¹A and –A₁ (Fig. 11) amino acids with five-membered cycles, respectively, both histidine (His) and tryptophan (Trp)  have been placed (fig. 12).

Instead of elements –²A, –A₂ (Fig. 12) in the bottom part of model side chains with six-membered cycles -phenylalanine (Phe) and tyrosine (Tyr) have been placed (fig. 13).

Fig. 12. An arrangement on the dodecahedron of the side chains of amino acids histidine and tryptophan.

Fig. 13. Position on the dodecahedron of amino acids of a subgroup 3

As a result we have received position on a dodecahedron of the subgroup 3 consisting of 8 amino acids

Оставшиеся 8 аминокислот канонического набора составили подгруппу 4

Only non-polar valine (Val) and isoleucine (Ile) may be the closest to the nonpolar alanine side chains. They have been placed instead of elements ¹B and B₁, respectively (fig. 14).

The side chains of larger size in this group - the polar aspartic and glutamic acids or non-polar glutamine and asparagine. Preference was given to the first pair, as antipodes to it may be only a polar arginine and lysine. Aspartic (Asp) and glutamic (Glu) acid have taken a place of elements ²B and B₂ (fig. 15).

Fig. 14. Position on the dodecahedron of amino acids valine and isoleucine.          

Fig. 15. An arrangement of the side chains of aspartic and glutamic acids on the dodecahedron.

Antipodes to a non-polar hydrophobic side chains valine and isoleucine are the non-polar hydrophilic side chains of asparagine (Asn) and glutamine (Glu). They have taken a place of the elements –¹B and –B₁, respectively (fig. 16).

Antipodes of aspartic and glutamic acids are arginine (Arg) and lysine (Lys), possessing alkaline properties. They have been substituted instead of elements –²B and –B₂ (fig. 17).

Fig. 16. Arrangement of amino acids asparagine and glutamine on the dodecahedron.

Fig. 17. An arrangement of amino acids of a subgroup 4 on the dodecahedron.

As a result, the position of the 8 side chains of amino acids of a subgroup 4 on the dodecahedron was obtained.

All 20 canonical amino acid side chains, located in accordance with the arrangement of subgroups 1-4 on a dodecahedron (Fig. 8, 9, 13, 17), is shown in Figure 18.

Subgroups 1 and 2 occupy the central area of the dodecahedron through which there passes a plane I. Subgroup 3, consisting of eight amino acids (pink circles) is located along the plane I. Subgroup 4 also consists of eight side chains (light blue circles) and is located along the plane III.

It should also be noted that the result of the construction the side chains of amino acids arranged in order of increasing molecular weight. At the top were the lightest side chains - Gly, Ala, Ser, Thr, etc., and at the bottom - the worst - His, Trp, Phe, Tyr. In the middle of the dodecahedron side chains were intermediate in mass between the lightest and heaviest.

This arrangement, however, was not done on purpose, but was the result of using the principles of anti-symmetry

Fig. 18. A complete system of the 20 canonical amino acid side chains on the dodecahedron.
Subgroups 1 and 2 - unpainted circles;
subgroup 3 - pink circles;
subgroup 4 – light blue circles.