Equilibrium in the Population of the Plants

Equilibrium in the Population of the Plants

DOI: 10.4018/978-1-5225-9651-6.ch001

Abstract

On the basis Mendel's experiments, a mathematical model is constructed that describes the results of these experiments in a wide range of parameters. This model is compared with the Hardy-Weinberg logistic model based only on probabilistic ideas about the presence of dominant and recessive alleles in the chromosomes of living organisms. 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 consists in the fact that with an increase in the number of generations all 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, seeking asymptotically to a stable isolated equilibrium position of the type of a multidimensional node. This newly discovered behavior of the dominant and recessive forms in the vicinity of the equilibrium position (true) differs significantly from the logistic equilibrium position in the Hardy-Weinberg principle, built without taking into account the real patterns in the plant population.
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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 works 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; Zhizhin, 2016, 2017, 2018). 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 free 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, 2004a). The combination of necessary mathematical conditions in the derivation of the Hardy -Weinberg formula does not alleviate this advantage.

Key Terms in this Chapter

Homozygous Individuals: The individuals that do not produce cleavage in the offspring.

Dominant Traits: The traits that predominate in the first generation.

Heterozygous Individuals: The individuals that produce cleavage in the offspring.

Monohybrid Crossing: A crossing in which the manifestation of only one trait is examined.

Polyhybrid Crossing: A crossing in which explores the manifestation of several signs.

Genotype: A set of genes of a given cell or organism.

Gene: A hereditary factor; functionally indivisible unit of genetic material; section of the DNA molecule encoding the primary structure of the polypeptide.

Recessive Traits: The traits that do not appear in the first generation.

Phenotype: The totality of all traits and properties of an individual that are formed in the process of interaction between its genetic structure and the external environment.

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