Genetic alphabets, unitary matrixes and quantum-algorithmic genetics

Unitary operators are the basis of all calculations in quantum computers and play an important role in quantum mechanics. The presented results show that matrix forms of representation of structured DNA and RNA alphabets can be considered as a set of sparse unitary matrices. These results are useful for the development of models of quantum-algorithmic genetics, from the point of view of which living organisms are quantum-information entities.


Introduction
Information molecules of DNA and RNA follow the principles of quantum mechanics. It can be assumed that genetic informatics uses quantum computing algorithms based on unitary (or orthogonal, in the case of real numbers) matrices. Any unitary matrix can be used as a logic gate in quantum computers [1]. The article presents some authors' results for the development of quantum-algorithmic genetics models [2][3][4][5][6][7][8].

The genetic code and binary oppositions
Why DNA alphabet is built by nature on the basis of only four letters: adenine A, guanine G, cytosine C and thymine T? Why is this alphabet based on these very simple molecules? Modern science does not know the answer to these questions. But it knows that this set of 4 "letters" has symmetric properties because it is endowed with three pairs of binary-opposition signs or indicators: 1) two letters are purines (A and G) and the other two are pyrimidines (C and T). In terms of these features the following can be taken: C = T = 0, A = G = 1; 2) two complementary letters (A and T) are connected by 2 hydrogen bonds, and other complementary letters (C and G) are connected by 3 hydrogen bonds. From the point of view of this feature, the following can be taken: A = T = 0, C = G = 1; 3) two letters are keto-molecules (G and T) and the other two are amino-molecules (A and C).
From the point of view of this feature, the following can be taken A = C = 0, G = T = 1. With regard to these three types of binary-oppositional traits in DNA letters any DNA text can be presented as a combination of three parallel messages in three binary languages. At the same time, these messages are mutually connected on the basis of the logical operation of modulo-2 addition used in quantum and conventional computers: the sum of modulo-2 addition of any two of these three binary representations of the DNA text always gives its third binary representation [4]. It is easy to check that each of the letters A, C, G, T is uniquely determined by any two of these three indicators. This allows presenting DNA alphabets of 4 nucleotides, 16 doublets, 64 triplets, etc. in a convenient form of square tables ( Fig. 1-3), in which lines and columns are numbered with binary numbers according to the following principle. The components of each column are binary numbered in accordance with the binary opposition signs "purine or pyrimidine" (for example, the CAG triplet and all other triplets in the same column are a bunch of "pyrimidine-purine-purine", and therefore this column is numbered 011). The components of each line are numbered with binary numbers in accordance with the signs of "keto-or amino-" (for example, the same CAG triplet and all other triplets in its line are a bunch of "amino-amino-keto", and therefore this line is numbered 001). In such tables ( Fig. 1-3) each of the members of the alphabets of 4 nucleotides, 16 doublets, 64 triplets, ... takes its individual place in a strict order  It is essential that these 3 separate genetic tables form a single tensor family of matrices based on the tensor (or Kronecker) product [2,4]. Indeed, the second tensor degree of (2x2)-nucleotide matrix [C, A; T, G] automatically gives the (4x4)-matrix of 16 doublets, and its third tensor degree gives the (8x8)matrix of 64 triplets with the same arrangement of components as in the tables in Fig. 1-3, which appear now as matrices. Thus, the structural organization of the system of DNA alphabets is connected with matrices and algebraic operation of the tensor product of matrices, which plays an important role in quantum mechanics and quantum computing in connection with the following: the state space of a multicomponent quantum system is the tensor product of the state spaces of its components [1, p. 71].
It is well known that the degeneracy of the genetic code is associated with the division of the whole set of 64 triplets by the nature into two equal subsets: 3 -32 triplets have "strong roots" (indicated in black in the matrixes in Fig. 1), i.e. start with 8 "strong" doublets AC, CC, CG, CT, GC, GG, GT, TC; -32 triplets have "weak roots" (white in Fig. 1), i.e. start with 8 "weak" doublets CA, AA, AG, AT, GA, TA, TG, TT. Code values of triplets with strong roots do not depend on the letters in the third position, but triplets with weak roots depend on the letters in the third position.
What is the location of these" black "and" white " members of the DNA alphabets in the genetic matrixes of 16 doublets and 64 triplets in Fig. 1? An unexpected phenomenological fact is their symmetrical arrangement in these matrices, built quite formally without mentioning the degeneracy of the genetic code. The symmetric properties of the resulting mosaics in these matrices are as follows: the left and right halves of the matrices are mirror-antisymmetric to each other; both quadrants along each diagonal are the same on the mosaic; the mosaic of each line has a meander character of the known Rademacher functions, which are a special case of the Walsh functions and contain only the numbers +1 and -1. Through this analogy with the Rademacher-Walsh functions from the theory of noise-immunity coding of information, the symbolic genomatrix from Fig. 1-3 are converted to numerical matrixes V and M with components +1 and -1, where components +1 (-1) represent black (white) doublets and triplets respectively (Fig. 4).  (Fig. 2) and 64 triplets (Fig. 1). Cells with numbers +1 (-1) correspond to cells with black (white) doublets and triplets in Fig. 1 and Fig.3. Each line of these numerical matrices coincides with one of the Rademacher-Walsh functions.

On unitary matrices
These matrices V and W (Fig. 4) represent the phenomenological features of the degeneracy of the genetic code. Below the properties of genetic matrices V and W will be analyzed more deeply. The analysis begins with (4x4)-matrix V, which turns out to be the sum of four sparse unitary (orthogonal) matrixes V = V0 + V1 + V2 + V3 (Fig. 5), for which the unitarity condition is fulfilled: V0*V0 T = I, V1*V1 T = I, V2*V2 T = I, V3*V3 T = I, where I is the unit matrix (this decomposition is not arbitrary, but is based on the principle of dyadic shifts from the theory of digital signal processing [2]).  Unitary transformations preserve the norms of vectors. The tensor product of two unitary matrixes always generates a new unitary matrix. According to the described result, the degeneracy of the genetic code is associated with sparse unitary matrices, which can serve as quantum gates in biological quantum computers.
A set of these unitary matrices V0, V1, V2 and V3 (Fig. 5) is closed with respect to multiplication. Their multiplication table (Fig. 4) coincides with the multiplication table of Cockle's split-quaternions (http://en.wikipedia.org/wiki/Split-quaternion), known in particular in connection with the Poincare disk model of Lobachevski geometry. It is to be added that the dyadic-shift decomposition of (8x8)-matrix W (Fig. 4) gives a set of 8 sparse unitary matrices, which is also closed with respect to multiplication and has a Cockle's bi-split quaternion multiplication table [4].
The reasons for the connection of structured DNA and RNA alphabets in their different matrix representations with Cockle's split-quaternions should be explained. It seems to be related to the connection of split-quaternions and Poincare disc model of Lobachevski geometry with known experimental data on Lobachevski geometry of the space of visual perception, which has resulted in some authors' model approaches to the brain on the basis of Clifford algebra of quantum computers (see details in [4]). Our results on the relationship between the genetic system and split-quaternions support the importance of developing algebra-geometric approaches to understanding the inherited properties of sensory systems and biological informatics in general.
The lines and columns of each of the unitary matrices V0, V1, V2 and V3 form an orthogonal system of functions that can be used for spectral analysis of physiological and other processes in their vector representations. The construction of each of these matrices to tensor powers gives new unitary matrixes of increased order with orthogonal systems of functions in their lines and columns.

Conclusions
The results presented in the article together with our other works [2][3][4][5][6][7][8][9] allow us to develop the elements of quantum-algorithmic genetics as one of the promising new directions of science. This scientific direction concerns also works of some other authors about quantum biology, which suggests that biological phenomena such as photosynthesis, enzyme catalysis, avian navigation or olfaction may use of a number of the non-trivial features of quantum mechanics, such as coherence, tunnelling and entanglement.