Higher Dimensions in the Theory of Heredity

Introduction

In 1865, Mendel made a presentation at the Society of Naturalists in Brynn about experiments on plant hybrids (Mendel, 1965). This presentation gave birth to the development of genetics as a science, although not immediately the content of the speech became of known to the scientific community and was appreciated (Gaisinovich, 1988). Understanding of its significant results took place in the struggle of opposing scientific trends and was accompanied by dramatic events in human relations. At different times, scientists have seen in these experiments different, sometimes opposite, results. More than 30 years after this speech, when the results work of Mendel were reopened and confirmed experimentally in the works of Correns (Correns, 1900), and De Vries (De Vries, 1904).

Over the past 150 years after the speech of Mendel, a chromosome theory of heredity was created, which gave a molecular explanation to the results of the experiments of Mendel (Weismann, 1885; Johannsen, 1933; Morgan, 1937; Koltsov, 1935; Chetverikov, 1926; Watson & Crick, 1953 a, b; Zhizhin, 2018, 2019 a). The same time after of the second discovery of Mendel's experiments, there appeared works in which it was noted that in Mendel's experiments there was a steady increase in the number of dominant alleles in populations, which indicated the absence of equilibrium positions in plant populations that obeyed Mendel's law.

In this regard, Yule (Yule, 1902) purely mathematically proved that in the case of absolutely random crossing in the population of heterozygous forms, there is an equilibrium between the number of dominant and recessive forms. Continued these studies Hardy (Hardy, 1908), who derived the formula for the distribution of genotypes in freely crossbreeding populations. Regardless of him and even earlier, Weinberg (Weinberg, 1908) established the same formula. This formula was called as principle the Hardy - Weinberg and became widespread. However, for the mathematical derivation of this formula, very strong assumptions are used: lack of choice in organisms, infinity of the population, accidental crossing of the population's organisms with each other, uniform distribution of male and female individuals, absence of mutation and genetic drift. The totality of these assumptions precludes the possibility of realizing such populations in nature. Therefore, this principle cannot be confirmed experimentally. Its only advantage is that it has equilibrium positions (a finite ratio of the numbers of dominant and recessive forms). However, these equilibrium positions are formed for any initial contents of these forms. Consequently, the set of equilibrium positions is a continuous manifold, and therefore they are asymptotically unstable, since any small perturbation can translate the system from one equilibrium position to another (Zhizhin, 1972, 2004 a). The combination of necessary mathematical conditions in the derivation of the Hardy -Weinberg formula does not alleviate this advantage.

In this Chapter, a mathematical analysis of the sequences of obtaining the values ​​of the number of phenotypes of organisms in Mendel's experiments was carried out. It is found that the patterns of inheritance of constant - differing sings, experimentally established by Mendel, are described by special algebraic relations, the basis of which are geometric progressions. Mendel and subsequent researchers did not pay attention to this. Naturally, Mendel could not trace the patterns of inheritance on experiments with a large number of generations. However, the mathematical model obtained in this Chapter (Zhizhin, 2019), accurately confirmed by Mendel's opaque in the field of conditions of their conduct, allows one to cram into the inheritance processes of sings with a large number of generations. A numerical and qualitative analysis of the equations of the mathematical model with an increase in the number of generations with an arbitrary number of pairs of constant - differing features was carried out.

There is shown that in the mathematical model of Mendel's experiments, based on real patterns of plant development, there are equilibrium positions between the dominant and recessive forms. It is shown that with an increase in the number of generations of the number all of dominant and recessive phenotypes of organisms with any number of sings quickly equalize and then synchronously (in the absence of death of organisms) increase together. In this case, the system asymptotically strive to achieve an isolated stable equilibrium position of the node type in a multidimensional space with coordinates in the form of quantities inversely proportional to the number of different phenotypes of organisms.