This chapter presents data about cyclic properties of the genetic code in its matrix forms of presentation. These cyclic properties concern cyclic changes of genetic Yin-Yang-matrices and their Yin-Yangalgebras (bipolar algebras) at many kinds of circular permutations of genetic elements in genetic matrices. These circular permutations lead to such reorganizations of the matrix form of presentation of the initial genetic Yin-Yang-algebra that arisen matrices serve as matrix forms of presentations of new Yin-Yang-algebras, as well. They are connected algorithmically with Hadamard matrices. New patterns and relations of symmetry are described. The discovered existence of a hierarchy of the cyclic changes of genetic Yin-Yang-algebras allows one to develop new algebraic models of cyclic processes in bioinformatics and in other related fields. These cycles of changes of the genetic 8-dimensional algebras and of their 8-dimensional numeric systems have many analogies with famous facts and doctrines of modern and ancient physiology, medicine, and so forth. This viewpoint proposes that the famous idea by Pythagoras (about organization of natural systems in accordance with harmony of numerical systems) should be combined with the idea of cyclic changes of Yin-Yang-numeric systems in considered cases. This second idea reminds of the ancient idea of cyclic changes in nature. From such algebraic-genetic viewpoint, the notion of biological time can be considered as a factor of coordinating these hierarchical ensembles of cyclic changes of the genetic multi-dimensional algebras.
Introduction And Background
This chapter continues an analysis of the genetic 8-dimensional Yin-Yang-algebra (bipolar algebra), which was described in Chapter 7. This analysis allows one to revelation of unknown properties of this genetic algebra and its possible applications for deeper understanding of genetic and physiological systems including inherited physiological cycles.
One of the directions, where the results of analysis of genetic Yin-Yang-algebras can be useful, is related to a creation of algebraic models of inherited physiological cycles and rhythms in organisms. The statement that biological organisms exist in accordance with cyclic processes of environment and with their own cyclic physiological processes is one of the most classical statements of biology and medicine from ancient times (see for example (Dubrov, 1989; Wright, 2002)). Many branches of medicine take into account the time of day specially, when diagnostic, pharmacological and therapeutic actions should be made for individuals. The set of this medical and biological knowledge is usually united under names of chrono-medicine and chrono-biology. Many diseases are connected with disturbances of natural biological rhythms in organisms. The problem of internal clocks of organisms, which participate in coordination of all interrelated processes of any organism, is one of the main physiological problems. But cyclic principles are essential for spatial organization of living bodies as well. Biological morphogenesis gives many examples of a cyclic symmetric repetition of separate spatial blocks in constructions of organism bodies (Figure 1). Such biological “cyclomerism” has been studied from viewpoints of Euclidean and non-Euclidean geometries for a long time (Petoukhov, 1989).
Some examples of inherited cyclic configurations in living matter: a leaf of fern, a cone, a shell of mollusk
Molecular biology deals with this problem of physiological rhythms and of cyclic re-combinations of molecular ensembles on the molecular level as well. Really, it is the well-known fact that in biological organisms proteins are disintegrated into amino acids and then they are re-built (are re-created) from amino acids again in a cyclic manner systematically. A half-life period (a duration of renovation of half of a set of molecules) for proteins of human organisms is approximately equal to 80 days in most cases; for proteins of the liver and blood plasma – 10 days; for the mucilaginous cover of bowels – 3-4 days; for insulin – 6-9 minutes (Aksenova, 1998, v. 2, p. 19). Such permanent rebuilding of proteins provides a permanent cyclic renovation of human organisms. Such cyclic processes at the molecular-genetic level are one of the parts of a hierarchical system of a huge number of interelated cycles in organisms. The phenomenon of repeated recombinations of molecular ensembles, which are carried out inside separate cycles, is one of the main problems of biological self-organization. This phenomenon draws additional attention to structural properties of recombinations and permutations of molecular elements of genetic code systems. We are studying these structural properties using a matrix language. One can mention here that in addition to cyclic renovation of proteins, heritable corporal forms of activity exist: cardio cycles, breath cycles, walking, run, crawling, swimming and so forth.
Do some structural connections of the genetic code systems with inherited physiological rhythms and with such cyclic processes exist? Matrix genetics proposes new mathematical data of structural analysis for a positive answer on the first question and for a creation of algebraic models of such hierarchical system of cyclic changes. These data were obtained on the basis of an analysis of the mentioned genetic 8-dimensional Yin-Yang-algebra. This algebra was revealed initially as a result of analysis of the genetic matrix [C A; U G](3), where the symbol in parentheses means the third Kronecker power and the symbols C, A, U, G mark nitrogenous bases of the genetic code (cytosine, adenine, uracil, guanine).
This genetic 8-dimensional Yin-Yang-algebra (bipolar algebra) was described in Chapter 7. This matrix algebra defines the system of 8-dimensional numbers YY with 8 real coordinates x0, x1, …, x7:YY