Резюме

Щербик В. В.¹, Бучацький Л. П.² Генетичний код – квантований тензор кривизни чотиривимірного риманового простору.

Пари амінокислот генетичного коду розділені на три незалежні сімейства, які представлені квадратичними формами. Коефіцієнти квадратичних форм дорівнюють сумі чисел ізоформ тРНК антикодонів спарених амінокислот. Парування амінокислот генетичного коду визначає всі 256 компонент квантованного тензора кривизни чотиривимірного ріманового простору, в якому незалежними компонентами є двадцять амінокислот генетичного коду. Алгебра Кліффорда квадратичних форм підмножин амінокислот є алгеброю гіперболічних підпросторів в чотиривимірному римановом просторі постійної кривизни, в якому відбувається синтез білка.

Ключові слова: генетичний код, алгебра Кліффорда, риманів простір, тензор кривизни.

Резюме

Щербик В. В., Бучацкий Л. П. Генетический код – квантованный тензор кривизны четырехмерного риманового пространства.

Пары аминокислот генетического кода разделены на три независимые семейства, которые представлены квадратичными формами. Коэффициенты квадратичных форм равны сумме чисел изоформ тРНК антикодонов спаренных аминокислот. Спаривание аминокислот генетического кода определяет все 256 компонент квантованного тензора кривизны четырехмерного риманового пространства, в котором независимыми компонентами являются двадцать аминокислот генетического кода. Алгебра Клиффорда квадратичных форм подмножеств аминокислот является алгеброй гиперболических подпространств в четырехмерном римановом пространстве постоянной кривизны, в котором происходит синтез белка.

Ключевые слова: генетический код, алгебра Клиффорда, риманово пространство, тензор кривизны.

Summary

Stcherbic V. V., Buchatsky L. P. The genetic code is quantized curvature tensor of four-dimensional Riemannian space.

Pairs of amino acids of the genetic code are divided into three separate families, represented by quadratic forms. The coefficients of quadratic forms equal the sum of the numbers of tRNA anticodons isoforms paired amino acids. Pairing of amino acids of the genetic code specifies all 256 components of quantized four-dimensional Riemannian curvature tensor of the space in which independent components are twenty amino acids of the genetic code. Clifford algebra of quadratic forms is the algebra of subsets of amino acids hyperbolic subspaces in four-dimensional Riemannian space of constant curvature, where protein synthesis occurs.

Keywords: the genetic code, Clifford algebra, Riemannian space, curvature tensor.

Рецензент: д.біол.н., провід.наук.співр. Л.Т. Міщенко

UDC 573.2

Киевский национальный университет имени Тараса Шевченко

Украина, 01601, г. Киев, ул. Владимирская, 64/13

Taras Shevchenko National University of Kyiv

64/13, Volodymyrska Street, City of Kyiv, Ukraine, 01601

¹stcherbic_v@bigmir.net

²irido1@bigmir.net

It is known that the genetic code, which expresses the dependence of the amino acids triplets mRNA degenerate. For amino acid sequences impossible uniquely reconstruct the sequence of triplets of nucleotides encoding mRNA.

We have previously suggested that the origin of the genetic code is closely connected with the Riemann geometry. Twenty amino acids of the genetic code exactly correspond to the number of independent components of the curvature tensor of four-dimensional space-time , which means that the degeneracy of the genetic code.

It is asserted that the genetic code is the curvature tensor of quantized four-dimensional Riemannian space in which protein synthesis occurs. But the genetic code contains only the 64 states of the triplet code, whereas the curvature tensor of four-dimensional Riemannian space has 256 components. If we make the pairing of amino acids belonging to different triplets of nucleotides of mRNA, the number of states of the genetic code can be doubled.

For this it is necessary to satisfy the condition Am1(i, j, k) ¹ Am2(k_π, j_π, i_π), where π is a permutation nucleotide triplet i, j, k. In particular, the codon-anticodon pairing amino acids satisfies Am1(i, j, k) ¹ Am2(k_π, j_π, i_π), so it is suitable for doubling the states of the genetic code .

We use the genetic code table, built on the basis of the codon- anticodon pairing amino acids. To re-doubling states of the genetic code, tRNA isoforms tables are used.

The essence of the quantization of the curvature tensor is that each independent component corresponded to a separate physical particle, not necessarily elementary. Since the curvature tensor has four index, then numbering 20, and physical needs of particles fifth index. Introduction 5 -dimensional space in such a clear way not possible in four-dimensional world. But it is possible to construct the equivalent of a 5-dimensional space by adding to the dynamic part of the curvature tensor of a stochastic part, the more that a Riemannian space in general is non-integrable .

Quantization of the curvature tensor is performed using the Clifford algebra of quadratic forms. Riemannian space is a source of complementary pairs of nucleotides mRNA-tRNA, which correspond to pairs of amino acids that form the hyperbolic space.

Hyperbolic space is the space with the signature [p+ p–], i. e. p intervals (planes) type z² = z₁² – z₂². Hyperbolic space has always even dimension. The double number z = z₁ + jz₂ is a Lorentz rotation. Collective motion of the 2p particles sets p Lorentz transformations for the space with the signature [p+ p–]. Degeneracy of the genetic code makes it impossible to separate the individual amino acids of their collective motion.

Table of dynamic genetic code is constructed in the form of pairs of amino acids which correspond to the codon-anticodon pairing nucleotides mRNA-tRNA. All codons occur once. Independent sets of amino acids form a hyperbolic space with signature [2+ 2–], [2+ 2–] and [3+ 3–]. Pairing of amino acids gives states doubling of the genetic code.

Table of stochastic genetic code is constructed in the form of pairs of amino acids which correspond to isoforms of the tRNA body Bos taurus (cow). Independent sets of amino acids form a hyperbolic space with signature [2+ 2–], [3+ 3–] and [4+ 4–]. Testing reveals that these signatures are not dependent upon the particular organism. Dynamic and stochastic genetic code contains 256 amino acids.

On the basis of transformations of sets of amino acids dynamic and stochastic genetic code can construct affine and pseudoaffine curvature tensors of four-dimensional space of protein synthesis.

Four-dimensional Riemannian curvature tensor space has two forms. The first form of the curvature tensor formed coaxial, twisted equally spirals mRNA-tRNA . The second form of the curvature tensor associated with the complementary pairing mRNA codon and tRNA anticodon.

Classical curvature tensor in four-dimensional Riemannian space has 112 zero elements , which corresponds to a quantized curvature tensor 112 different pairs of amino acids in four pages.

The genetic code is very stable. Therefore, the curvature tensor of quantized four-dimensional Riemannian space, which is the genetic code, can only be a space of constant curvature tensor . The operator of the curvature tensor R4 may depend only on the components of the metric tensor. In the formula R4 are grouped hyperbolic subspaces in common four-dimensional Riemannian space of amino acids of the genetic code.

Conclusions. The genetic code is the curvature tensor of quantized four-dimensional Riemannian space of protein synthesis. Pairing of amino acids of the genetic code specifies all 256 components of the curvature tensor. Quantized representation of twenty independent components of the curvature tensor is twenty amino acids of the genetic code.