Sergey Petoukhov

The system-resonance approach in modeling genetic structures.

The founder of the theory of resonance in structural chemistry Linus Pauling established the importance of resonance patterns in organization of living systems. Any living organism is a great chorus of coordinated oscillatory processes. From the formal point of view, biological organism is an oscillatory system with a great number of degrees of freedom. Such systems are studied in the theory of oscillations using matrix mathematics of their resonance characteristics. This study is devoted to a new approach for modeling genetically inherited structures and processes in living organisms using mathematical tools of the theory of resonances. This approach reveals hidden relationships in a number of genetic phenomena and gives rise to a new class of bio-mathematical models, which contribute to a convergence of biology with physics and informatics. In addition some relationships of molecular-genetic ensembles with mathematics of noise-immunity coding of information in modern communications technology are shown. Perspectives of applications of the phenomena of vibrational mechanics for modeling in biology are discussed.

I-Ching, dyadic groups of binary numbers and the geno-logic coding in living bodies

The ancient Chinese book I-Ching was written a few thousand years ago. It introduces the system of symbols Yin and Yang (equivalents of 0 and 1). It had a powerful impact on culture, medicine and science of ancient China and several other countries. From the modern standpoint, I-Ching declares the importance of dyadic groups of binary numbers for the Nature. The system of I-Ching is represented by the tables with dyadic groups of 4 bigrams, 8 trigrams and 64 hexagrams, which were declared as fundamental archetypes of the Nature. The ancient Chinese did not know about the genetic code of protein sequences of amino acids but this code is organized in accordance with the I-Ching: in particularly, the genetic code is constructed on DNA molecules using 4 nitrogenous bases, 16 doublets, and 64 triplets. The article also describes the usage of dyadic groups as a foundation of the bio-mathematical doctrine of the geno-logic code, which exists in parallel with the known genetic code of amino acids but serves for a different goal: to code the inherited algorithmic processes using the logical holography and the spectral logic of systems of genetic Boolean functions. Some relations of this doctrine with the I-Ching are discussed. In addition, the ratios of musical harmony that can be revealed in the parameters of DNA structure are also represented in the I-Ching book.

The Genetic Coding, United-Hypercomplex Numbers and Artificial Intelligence

Scientists try to reproduce in devices of artificial intelligence intellectual properties of living organisms, which are connected with the genetic code system. This article is devoted to the study and modeling of the genetic system on the basis of mathematical formalisms, which are used in digital devices of artificial intelligence and technology of noise-immunity coding of information. The genetic code of amino acid sequences in proteins does not allow understanding and modeling of inherited processes such as inborn coordinated motions of living bodies, innate principles of sensory information processing, quasi-holographic properties, etc. To be able to model these phenomena, the concept of geno-logical coding, which is connected with logical functions and Boolean algebra, is put forward. Structured alphabets of DNA in their matrix form of representations are connected with dyadic groups of binary numbers and a new type of systems of multidimensional numbers. This type generalizes systems of complex numbers and hypercomplex numbers, which serve as the basis of mathematical natural sciences and many technologies. The new systems are called in a general case as “systems of united-hypercomplex numbers”. They can be widely used in models of multi-parametrical systems in the field of algebraic biology, artificial life, devices of biological inspired artificial intelligence, etc.

THE GENETIC CODE, HADAMARD MATRICES AND ALGEBRAIC BIOLOGY

Algebraic theory of coding is one of modern fields of applications of algebra. This theory uses matrix algebra intensively. This paper is devoted to an application of Kroneckers matrix forms of presentations of the genetic code for algebraic analysis of a basic scheme of degeneracy of the genetic code. Similar matrix forms are utilized in the theory of signal processing and encoding. The Kronecker family of the genetic matrices is investigated, which is based on the genetic matrix [C A; U G], where C, A, U, G are the letters of the genetic alphabet. This matrix in the third Kronecker power is the (8*8)-matrix, which contains all 64 genetic triplets in a strict order with a natural binary numeration of the triplets by numbers from 0 to 63. Peculiarities of the basic scheme of degeneracy of the genetic code are reflected in the symmetrical black-and-white mosaic of this genetic (8*8)-matrix. This mosaic matrix is connected algorithmically with Hadamard matrices unexpectedly, which are famous in the theory of signal processing and encoding, spectral analysis, quantum mechanics and quantum computers. Furthermore, many kinds of cyclic permutations of genetic elements lead to reconstruction of initial Hadamard matrices into new Hadamard matrices unexpectedly. This demonstrates that matrix algebra is one of promising instruments and of adequate languages in bioinformatics and algebraic biology.

New Symmetries and Fractal-Like Structures in the Genetic Coding System

The achievements of molecular genetics and bioinformatics lead to significant changes in technological, medical and many other areas of our lives. This article is devoted to new results of study of structural organization of genetic information in living organisms. A new class of symmetries and fractal-like patterns in long DNA-texts is represented in addition to two Chargaff’s parity rules, which played an important role in development of genetics and bioinformatics. Our results provide new approaches for modeling genetic informatics from viewpoints of quantum informatics and theory of dynamic chaos.

Triply Stochastic Cubes Associated with Genetic Code Numerical Mappings

Knowledge about genetic coding systems are useful for computer science, engineering and education. In this paper we derive triply stochastic cubes associated with the triplet genetic code numerical mappings. We also demonstrate the symmetrical patterns between the entries of the cubes and DNA molar concentration accumulation via an arithmetic sequence. The stochastic cubes based on genetic code were derived by using three kinds of chemically determined equivalences. We have shown that at each stage (Nth step) of matrix evolution, hydrogen bonds expansion is triply stochastic and its accumulation is governed by an arithmetic sequence with a common difference of total number of hydrogen bonds of 5N; the pyrimidines/purines ring expansion is triply stochastic and its accumulation is governed by an arithmetic sequence with a common difference of total number of rings of 3N; and the amino-mutating absence/present expansion is also triply stochastic and its accumulation is governed by an arithmetic sequence with a common difference of total number of amino-mutating of 1N. Data about the genetic stochastic matrices/cubes associated with the genetic codes can lead to new understanding of genetic code systems, to new effective algorithms of information processing which has a perspective to be applied for modeling mutual communication among different parts of the genetic ensemble.

On symmetries, resonances and photonic crystals in morphogenesis

Biological symmetries, theories of the morphogenetic field, resonant interactions and the role of photons in morphogenetic processes represented the main fields of interest of Lev Beloussov and his followers. This review article includes some results of our study on the important role of resonances and photonic crystals in genetic informatics. Mathematical formalisms of differential Riemannian geometry and tensor analysis are used for modeling inherited curved surfaces in biomorphology and for understanding conformal bio-symmetries connected with the networks of curvature lines of surfaces. Notions of a morpho-resonance field as one of variants of morphogenetic fields are discussed. The connection of the golden section with the Fibonacci matrix of growth used in morphogenetic models of phyllotaxis is shown. Photonic crystals are considered as important participants of organisation of molecular-genetic informatics.

A Mathematical Proof of Double Helix DNA to Reverse Transcription RNA for Bioinformatics

This paper presents a mathematical proof of deoxyribose nucleic acid (DNA) to ribonucleic acid (RNA) based on the block circulant Jacket matrix (BCJM) characteristics, which is used to develop a bioinformatics for the molecular communications. The DNA matrix decomposition is the form of the Kronecker product of identity and Hadamard matrices with pair complementarity. The RNA 4 by 4 genetic matrix is the anti-symmetric pair complementary of the core kernel. The variants of kernel of the Kronecker families are produced by permutations of the four letters C, A, U, G on positions in the matrix. Thus, we get 6 subset pattern of block circulant matrix, 6 upper-lower block symmetric matrix and 6 left-right block symmetric matrix. This decomposition of DNA to RNA leads very clearly to the Kronecker product of the symmetrical genetic matrices.

Matrices in Improvement of Systems of Artificial Intelligence and Education of Specialists

This article is devoted to a significant role of matrices in digital signal processing, systems of artificial intelligence and mathematical natural sciences in the whole. The study of the world of matrices is going on intensively all over the world and constantly brings useful and unexpected results. Some of these results are presented in this paper. Special attention is paid to Hadamard matrices and some their modifications and extensions, which are important for developing systems of artificial intelligence and studying the genetic code. Training courses for specialists in many fields of science should be constantly updated with new knowledge about matrices and their practical applications.

The Matrix Method of Representation, Analysis and Classification of Long Genetic Sequences

The article is devoted to a matrix method of comparative analysis of long nucleotide sequences by means of presenting each sequence in the form of three digital binary sequences. This method uses a set of symmetries of biochemical attributes of nucleotides. It also uses the possibility of presentation of every whole set of N-mers as one of the members of a Kronecker family of genetic matrices. With this method, a long nucleotide sequence can be visually represented as an individual fractal-like mosaic or another regular mosaic of binary type. In contrast to natural nucleotide sequences, artificial random sequences give non-regular patterns. Examples of binary mosaics of long nucleotide sequences are shown, including cases of human chromosomes and penicillins. The obtained results are then discussed.