Hyperbolic rules of the cooperative organization of eukaryotic and prokaryotic genomes

Universal dynamical function behind all genetic codes: P-adic attractor dynamical model

The genetic code is a map which gives the correspondence between codons in DNA and amino acids. In the attractor dynamical model (ADM), genetic codes can be described as the sets of the cyclic attractors of discrete dynamical systems - the iterations of functions acting in the ring of 2-adic integers Z2. This ring arises from representation of nucleotides by binary vectors and hence codons by triples of binary vectors. We construct a Universal Function B such that the dynamical functions for all known genetic codes can be obtained from B by simple transformations on the set of codon cycles - the "Addition" and "Division" operations. ADM can be employed for study of phylogenetic dynamics of genetic codes. One can speculate that the "common ancestor genetic code" was caused by B. We remark that this function has 24 cyclic attractors which distribution coincides with the distribution for the hypothetical pre-LUCA code. This coupling of the Universal Function with the pre-LUCA code assigns the genetic codes evolution perspective to ADM. All genetic codes are generated from B through the special chains of the "Addition" and "Division" operations. The challenging problem is to assign the biological meaning to these mathematical operations.

Generation of genetic codes with 2-adic codon algebra and adaptive dynamics

This is a brief review on modeling genetic codes with the aid of 2-adic dynamical systems. In this model amino acids are encoded by the attractors of such dynamical systems. Each genetic code is coupled to the special class of 2-adic dynamics. We consider the discrete dynamical systems, These are the iterations of a function F:Z2→Z2, where Z2 is the ring of 2-adic numbers (2-adic tree). A genetic code is characterized by the set of attractors of a function belonging to the code generating functional class. The main mathematical problem is to reduce degeneration of dynamic representation and select the optimal generating function. Here optimality can be treated in many ways. One possibility is to consider the Lipschitz functions playing the crucial role in general theory of iterations. Then we minimize the Lip-constant. The main issue is to find the proper biological interpretation of code-functions. One can speculate that the evolution of the genetic codes can be described in information space of the nucleotide-strings endowed with ultrametric (treelike) geometry. A code-function is a fitness function; the solutions of the genetic code optimization problem are attractors of the code-function. We illustrate this approach by generation of the standard nuclear and (vertebrate) mitochondrial genetics codes.

The stochastic organization of genomes and the doctrine of energy-information evolution based on bio-antenna arrays

The article is devoted to the possibilities of considering the evolution of biological systems in connection with the unique emergent properties of antenna arrays, that is, systems of mutually matched antennas widely used in technology. Materials are presented in favor of the proposition that the evolution of biosystems can be formally considered as the evolution of systems of bio-antenna arrays and their energy-information wave activity, which participates in biological computation and contributes to the unification of body parts into a coherent whole. The use of digital antenna arrays in technology is based on their tensor-matrix theory. The author discovers a structural analogy of this theory with the tensor-matrix features of genetic coding systems, as well as algebraic modeling of the universal rules for the stochastic DNA organization of the genomes of higher and lower organisms. This analogy is just one of the facts presented in the article in favor of the usefulness of borrowing knowledge from modern antenna technology to consider the evolution of biosystems. The described new approach may exist along with other known approaches in evolutionary biology.

Spectral Decomposition of Mappings of Molecular Genetic Information in the System Basis of Single Nucleotide Functions

This paper presents and visualizes examples of large amounts of genetic information using a new class of cognitive computer graphics algorithms. These algorithms are related to the semiotics of perception and allow the interpretation of those properties of nucleotide sequences that are difficult to perceive by simple reading or by standard means of statistical analysis. This article summarizes previously presented algorithms for visualizing long nucleic acids based on the primary Hadamard–Walsh function system. The described methods allow us to produce one-dimensional mappings of nucleic acids by levels corresponding to their scale-integral physicochemical parameters and construct a spectral decomposition of the nucleotide composition. An example of the spectral decomposition of parametric representations of molecular genetic structures is given. In addition, a multiscale composition of genetic functional mappings visualizing the structural features of nucleic acids is discussed.

Algebraic harmony and probabilities in genomes. Long-range coherence in quantum code biology

According to the founders of quantum mechanics and quantum biology P. Jordan and E. Schrödinger, the main difference between living and inanimate objects is the dictatorial influence of genetic molecules on the whole living organism. Code biology can make a valuable contribution to understanding this dictatorial influence of genetic molecules whose ensemble is endowed with many interconnected alphabets and codes. The paper is devoted to probability rules of nucleotide sequences of single-stranded DNA in eukaryotic and prokaryotic genomes. These rules are connected with n-plets alphabets of DNA whose nucleotide sequences are considered as bunches of many parallel texts written in interconnected n-plets alphabets. The rules draw attention to genomic phenomena of special tetragroupings of n-plets and new genomic symmetries. A generalization of the second Chargaff's rule is described. They show the existence of long-range coherence in genomic DNA sequences and reveal new connections of structural features of genomic sequences with formalisms of quantum mechanics and quantum informatics. The author supposes that the received results are related to the known vibration-resonance theory of G. Frohlich about long-range coherence in biological systems, that is, about collective quantum effects there. The possible influence of the described genetiс probability phenomena on the genetically inherited physiological structures is noted and discussed.

Harmonic Progression in Bioinformatics and Recurrent Series in Inherited Biostructures

The article is devoted to the study of new approaches to the development of mathematical models in genetic biomechanics, which studies the structural relationships of the genetic coding system with genetically inherited biological forms. More specifically, we are talking about models based on the recurrent harmonic progression whose connection with the information sequences of DNA molecules in the genomes of higher and lower organisms was recently revealed. In particular, the article describes previously unknown connections of the function of natural logarithms with the structures of the molecular genetic system, which allow modelling the main psychophysical logarithmic law by Weber-Fechner and also many other logarithmic structures in genetically inherited biological systems. In physics, the harmonic progression is traditionally considered, first of all, as related to standing waves in resonators. Our results are correlated with Frohlich’s vibration-resonant theory about collective quantum effects and long-range communication in biological systems.

Non-Euclidean Biosymmetries and Algebraic Harmony in Genomes of Higher and Lower Organisms

The article is devoted to the study of the relationship of non-Euclidean symmetries in inherited biostructures with algebraic features of information nucleotide sequences in DNA molecules in the genomes of eukaryotes and prokaryotes. These genomic sequences obey the universal hyperbolic rules of the oligomer cooperative organization, which are associated with the harmonic progression 1/1, 1/2, 1/3,.., 1/n. The progression has long been known and studied in various branches of mathematics and physics. Now it has manifested itself in genetic informatics. The performed analysis of the harmonic progression revealed its connection with the cross-ratio, which is the main invariant of projective geometry. This connection consists in the fact that the magnitude of the cross-ratio is the same and is equal to 4/3 for any four adjacent members of this progression. The long DNA nucleotide sequences have fractal-like structure with so called epi-chains, whose structures are also related to the harmonic progression and the projective-geometrical symmetries. The received results are related additionally to a consideration of DNA double helix as helical antenna. This fact of the connection of genetic informatics with the main invariant of projective geometry can be used to explain the implementation of some non-Euclidean symmetries in genetically inherited structures of living bodies.

Modeling inherited physiological structures based on hyperbolic numbers

The article is devoted to using the mathematics of multi-dimensional hyperbolic numbers and their matrix representations for modeling of different inherited biosystems, which are parts of the holistic body. The list of considered models includes phyllotaxis sequences; Punnet squares of the Mendelian genetics; the main psychophysical Weber-Fechner law, and some others. This modeling approach reveals hidden structural interconnections among different biosystems and leads a deeper understanding of their commonality related to the commonality of their genetic basis. At the same time, it shows the structural connection of inherited biosystems with formalisms of the theory of resonances of oscillatory systems with many degrees of freedom. This complements works of many authors - from antiquity to the present day - on the importance of coordinated oscillations and resonances in the life of living organisms. The modeling approach based on hyperbolic numbers and their bisymmetric matrices reveals that the structural commonality of different inherited biosystems is related to the harmonic progression 1/n and the harmonic mean, which are long known in arts and culture. They are used in the Pythagorean teaching on the musical harmony and the aesthetics of proportions. Besides, the using - for biosystems modeling - of multi-dimensional hypercomplex numbers, which are one of the main tools in modern mathematical natural sciences, opens a bridge between these sciences and biology for their mutual enrichment.

Hyperbolic rules of the cooperative organization of eukaryotic and prokaryotic genomes

The author's method of oligomer sums for analysis of oligomer compositions of eukaryotic and prokaryotic genomes is described. The use of this method revealed the existence of general rules for the cooperative oligomeric organization of a wide list of genomes. These rules are called hyperbolic because they are associated with hyperbolic sequences including the harmonic progression 1, 1/2, 1/3, .., 1/n. These rules are demonstrated by examples of quantitative analysis of many genomes from the human genome to the genomes of archaea and bacteria. The hyperbolic (harmonic) rules, speaking about the existence of algebraic invariants in full genomic sequences, are considered as candidates for the role of universal rules for the cooperative organization of genomes. The results concerns additionally the problem of the origin of life. The described phenomenological results were obtained as consequences of the previously published author's quantum-information model of long DNA sequences. The oligomer sums method was also applied to the analysis of long genes and viruses including the COVID-19 virus; this revealed, in characteristics of many of them, the phenomenon of such rhythmically repeating deviations from model hyperbolic sequences, which are associated with DNA triplets. In addition, an application of the oligomer sums method is shown to the analysis of amino acid sequences in long proteins like the protein Titin. The topics of the algebraic harmony in living bodies and of the quantum-information approach in biology are discussed.