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# Gonosomal Algebra

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## Abstract

We introduce the gonosomal algebra. Gonosomal algebra extend the evolution algebra of the bisexual population (EABP) defined by Ladra and Rozikov. We show that gonosomal algebras can represent algebraically a wide variety of sex determination systems observed in bisexual populations. We illustrate this by about twenty genetic examples, most of these examples cannot be represented by an EABP. We give seven algebraic constructions of gonosomal algebras, each is illustrated by genetic examples. We show that unlike the EABP gonosomal algebras are not dibaric. We approach the existence of dibaric function and idempotent in gonosomal algebras.

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... In such systems the accompanying algebra will play the role of the sex differentiation algebra. Then an ACM will play the role of an evolution algebra of such biological systems (see [8,10,13,15,17,18] for different kinds of evolution algebras). ...
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Article
The purpose of these notes is to give a rather complete presentation of the mathematical theory of algebras in genetics and to discuss in detail many applications to concrete genetic situations. Historically, the subject has its origin in several papers of Etherington in 1939- 1941. Fundamental contributions have been given by Schafer, Gonshor, Holgate, Reiers¢l, Heuch, and Abraham. At the moment there exist about forty papers in this field, one survey article by Monique Bertrand from 1966 based on four papers of Etherington, a paper by Schafer and Gonshor's first paper. Furthermore Ballonoff in the third section of his book "Genetics and Social Structure" has included four papers by Etherington and Reiers¢l's paper. Apparently a complete review, in par­ ticular one comprising more recent results was lacking, and it was difficult for students to enter this field of research. I started to write these notes in spring 1978. A first german version was finished at the end of that year. Further revision and translation required another year. I hope that the notes in their present state provide a reasonable review and that they will facilitate access to this field. I am especially grateful to Professor K. -P. Hadeler and Professor P. Holgate for reading the manuscript and giving essential comments to all versions of the text. I am also very grateful to Dr. I. Heuch for many discussions during and after his stay in TUbingen. I wish to thank Dr. V. M.
Article
We consider an evolution algebra which corresponds to a bisexual population with a set of females partitioned into finitely many different types and the males having only one type. We study basic properties of the algebra. This algebra is commutative (and hence flexible), not associative and not necessarily power-associative, in general. We prove that being alternative is equivalent to being associative. We find conditions to be an associative, a fourth power-associative, or a nilpotent algebra. We also prove that if the algebra is not alternative then to be power-associative is equivalent to be Jordan. Moreover it is not unital. In a general case, we describe the full set of idempotent elements and the full set of absolute nilpotent elements. The set of all operators of left (right) multiplications is described. Under some conditions it is proved that the corresponding algebra is centroidal. Moreover the classification of 2-dimensional and some 3-dimensional algebras are obtained.
Article
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The zygotic algebra 3 for a finite number of sex-linked characters with arbitrary segregation rates is defined in two equivalent ways. A sufficient condition for the existence and uniqueness of non-trivial idempotents in a baric ideal B of 3 is given. A convergence theorem for the sequence of plenary powers of elements of B of unit weight is proved. In the case of Mendelian segregation in the male sex the conditions for uniqueness of idempotents and for convergence are the same for symmetric inheritance and sex-linked inheritance. The special cases of Mendelian and additive segregation rates in the females are discussed in greater detail.
Article
Etherington introduced certain algebraic methods into the study of population genetics ( 6 ). It was noted that algebras arising in genetic systems tend to have certain abstract properties and that these can be used to give elegant proofs of some classical stability theorems in population genetics ( 4 , 5 , 9 , 10 ).
Article
Ever since Mendel promulgated his famous laws, probability theory and statistics have played an important role in the study of heredity (9). Etherington introduced some concepts of modern algebra when he showed how a nonassociative algebra can be made to correspond to a given genetic system (1, 4). The fact that many of these algebras have common properties has led to their study from a purely abstract point of view (2, 3, 5, 6, 11, 12). Furthermore, the techniques of algebra give new ways of attacking problems in genetics such as that of stability.
Article
Most work in genetic algebras has been concerned with inheritance which is symmetric with respect to sex, in that the characters studied are determined by genes located at autosomal loci, and it is assumed that the segregation pattern is the same in males and females. When asymmetric situations are studied, the development of the theory is complicated by the higher dimensions of the algebras, and by a feature to which Etherington (3, p. 40) drew attention, namely the fact that the passage from the gametic to the zygotic algebra no longer quite corresponds to the process of duplication, as it does in the symmetric case. Etherington gave some results for the gametic and zygotic algebras of a single sex linked diallelic locus, and its properties were discussed further by Gonshor (4, p. 44). In a second paper (5, p. 334) Gonshor studied sex linkage in the case of multiple alleles, choosing a canonical basis which exhibited very clearly the multiplication table and ideal structure of the algebra. His treatment from the statement of the multiplication table in terms of the natural basis to its expression in terms of a canonical basis, is repeated in the displayed relations (4)–(8) below, for completeness and to establish the present notation.(Received August 02 1969)
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We shall extend some of the results of (7) to the case of multiple alleles, our primary concern being that of polyploidy combined with multiple alleles. Generalisations often tend to make the computations more involved as is expected. Fortunately here, the attempt to generalise has led to a new method which not only handles the case of multiple alleles, but is an improvement over the method used in (7) for the special case of polyploidy with two alleles. This method which consists essentially of expressing certain elements of the algebra in a so-called “ factored ” form, gives greater insight into the structure of a polyploidy algebra, and avoids a great deal of the computation with binomial coefficients, e.g. see (7), p. 46.
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Article
The joint allele frequency distribution is studied for any number of partially and completely sex-linked loci in an infinite population. The model assumes random mating and no selection or mutation, but includes arbitrary linkage distributions. The mathematical treatment is simplified by means of differential operators applied to elements in a linear algebra. Procedures are given which lead to explicit expressions for the frequencies after n generations. It is shown that the alleles at different loci are randomly distributed when nx.
Article
The zygotic algebra for sex linkage with multiple alleles contains an idealB which is a baric algebra. This ideal possesses at least one idempotent with non negative coefficients. For the mutation case and the case of simple Mendelian inheritance in the female sex convergence theorems are proved for the sequence of plenary powers of a normalized element. For these two cases it is shown that the idealB is a special train algebra.
Book
In this section we provide a short introduction to “algebras in genetics”. We explain how algebras arise in population genetics and we construct some examples of gametic algebras. These examples will be reconsidered and discussed in a wider frame-work in section 7. We introduce zygotic and copular algebras and we discuss the construction of these algebras from gametic algebras. The exposition will reveal various elementary though fundamental properties of these algebras.
Article
We introduce an (evolution) algebra identifying the coefficients of inheritance of a bisexual population as the structure constants of the algebra. The basic properties of the algebra are studied. We prove that this algebra is commutative (and hence flexible), not associative and not necessarily power associative. We show that the evolution algebra of the bisexual population is not a baric algebra, but a dibaric algebra and hence its square is baric. Moreover, we show that the algebra is a Banach algebra. The set of all derivations of the evolution algebra is described. We find necessary conditions for a state of the population to be a fixed point or a zero point of the evolution operator which corresponds to the evolution algebra. We also establish upper estimate of the limit points set for trajectories of the evolution operator. Using the necessary conditions we give a detailed analysis of a special case of the evolution algebra (bisexual population of which has a preference on type "1" of females and males). For such a special case we describe the full set of idempotent elements and the full set of absolute nilpotent elements. Comment: 21 pages
Article
Parthenogenesis is a phenomenon of undoubted biological interest which leads to the production of living young in many types of animals, as well as in plants. White (1977) has estimated that the proportion of animal 'species' which reproduce exclusively by parthenogenesis, and are therefore all female, is of the order of one in a thousand. Parthenogenesis may initiate early embryonic development in mammals, and its lack of success in this class poses some fundamental and as yet unresolved problems regarding the significance of fertilisation in the physiology of reproduction and embryonic development. Spontaneous and induced parthenogenesis in various species are reviewed in this article.
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Article
In fish, an amazing variety of sex determination mechanisms are known, ranging from hermaphroditism to gonochorism and from environmental to genetic sex determination. This makes fish especially suited for studying sex determination from the evolutionary point of view. In several fish groups, different sex determination mechanisms are found in closely related species, and evolution of this process is still ongoing in recent organisms. The medaka (Oryzias latipes) has an XY-XX genetic sex determination system. The Y-chromosome in this species is at an early stage of evolution. The molecular differences between X and Y are only very subtle and the Y-specific segment is very small. The sex-determining region has accumulated duplicated sequences from elsewhere in the genome, leading to recombinational isolation. The region contains a candidate for the male sex-determining gene named dmrt1bY. This gene arose through duplication of an autosomal chromosome fragment of linkage group 9. While all other genes degenerated, dmrt1bY is the only functional gene in the Y-specific region. The duplication leading to dmrt1bY occurred recently during evolution of the genus Oryzias. This suggests that different genes might be the master sex-determining gene in other fish.