Contribuições à teoria da genética em populações

Abstract

1) The equilibrium in populations, initially composed of various genotypes depends essentially from the mode of reproduction the relative viability and fertility of the competing genotypes, and the initial frequencies. 2) We have to distinguish two main types of sexual reproduction: Cross fertilization or random mating where each individual has equal chances to be fertilized by any individual of the population, and Self-fertilization where each individum is automatically selfed. Finally we encounter cases of a mixed reproduction where there is no free intercrossing nor an absolute selfing. 3) Populations, heterozygous for one pair of genes and without selection. a) In cross fertilized populations equilibrium is reached in acordance with the Hardy-Wéinberg rule, in the first generation: Inicial : AA /u + Aa /v + aa/u = 1 Equilibrio ( u + v/2)² + 2 ( u + v/2 ) (w + v/2) + ( w + v/2)²= 1 Equilibrio ( u + v/2)² + 2 (u + v/2) ( w + v/2) + ( w + v/2)² = p o² + 2. po.qo + q o² = 1 b) In self fertilized populations equilibrium will be reached thermiticialy by the complete elimination of all heterozigotes, but a, sufficient approach to equilibrium may . be reached quickly: Frequência mº Geração Final AA u + 2m-1/2m+1 v u + 1/2 v Aa 2/ 2m + 1 v - aa w + 2m-1 / 2m +1 v w + 1/2 v c) In populations with mixed reproduction, the equilibrium depends upon the relative frequencies of crossing and selfing* and the final values for some cases are given in Quadro I. 4) Survival Indices - Mathematical formulas are simplified if we introduce special survival indices for the homozygotes in relation to the survival of the heterozygotes. If we designate the absolve survival values of the three genotypes AA, Aa and aa, by x, y and z. we may give the following mathematical definition: x [ A A] : y [ Aa] : z [ aa] = x/y [ A A] : [ Aa] : z/ y [aa] = R A [ AA] : 1 [Aa] : Ra [aa] Thus the value of R can have any value between zero, i. e. complete elimination of homozygotés, and infinite, i. e. complete elimination of heterozygotes. The terms (1 -K) of Haldane and (1 -S) or W of Wright do not have this mathematical property, and may have only values from zero to one Evidently, in accordance with the nature of the elimina tory process, we shall have to subdivide these indices of total survival R into at least two main components: the survival during the vegetative phase and the survival of gametes during the reproduction phase. These two componentes are united by the following equation R = RA . RR 5) Populations with random mating and selection. a) The final equilibrium, which is independent from the Initial gene or genotype frequencies, is reached when the genes and the zygotic genotypes are present in the following proportions: (Quadro 2). Po/ qo = 1- Ro/ -R A [AA] ( 0,5 - Ra). R AV [Aa] = 4. ( 0,5 - Ra) . (0.5 -R A) [aa] ( 0,5 - R A)² . Rav b) Formulas are given which permit the calculus of frequencies in intermediate generations. 6) Discussion of special cases. a) Heterotic Genes. (Quadro 3 and Fig. 1). Ra < 1; Ra < 1 Inicial : Final : p (A)/q(a) -> 1-ra/1-ra = positivo/zero = infinito At equilibrium both genes remain present in the population, and their frequency will be increased or decreased whenever the initial frequencies were bigger or smaller than the values expected at equilibrium. b) Letais or semiletal, recessives genes. (Quadro 4 and Fig. 2). Rª ; Oz Ra < 1 Inicial : Equilibrio biológico Equilíbrio Matemático pa(A)/q(a) -> positivo /zero -> 1- Rq/ 1-Ra = positivo/negativo Equilibrium will be reached when the gene which causes the reduction of viability or fertility of homozigotes, is eliminated from the population. c) Semiletals, and parcially dominant genes. (Quadro 5 and Fig. 3). Rª ; Oz Ra < 1 Inicial : Equilibrio biológico Equilíbrio Matemático pa(A)/q(a) -> positivo /zero -> 1- Rq/ 1-Ra = positivo/negativo d) Incompatible Genes. Ra > 1 ; Ra > 1; Ra > Ra Equílibrio/biológico p (A)/ q(a) -> positivo/zero Equilibrio matemático -> positivo/ zero -> zero/negativo -> 1-Ra/1 - Ra = negativo/negativo In these two cases we have to distinguish between the biological and the mathematical equilibrium. The first will be reached when one of the genes has a frequency equal to zero, While negative values can have a mathematical meaning only. Thus biological equilibrium will be reached with the elimination of the gene which causes a relatively lower viability of the respective homozygotes. However we must note still one exception in the last case of incompatible genes: When both genes have the same initial frequency and when the respective survival values are also equal and larger than one, both the genes will remain in the population, and the homozygotes will become more frequent than the heterozygotes, thus furnishing an isolating mecha-nisme. e) Two special cases were discussed in some detail: differential elimination in the two sexes, and gametophite competition. (Quadro 6 e 7 e Pig. 4 a 6). 7) Populations with self-fertilization. a) Equilibrium will be reached when the three genotypes are present in the following proportions. (Quadro 8). [AA] ( 0,5 - Ra). R AV [Aa] = 4. ( 0,5 - Ra) . (0.5 -R A) [aa] ( 0,5 - R A) . Rav b) Formulas are given which permit the determination of genotype frequencies in intermediate generations. 8) Special cases We may say that, in a general way, the diferences (1 -R) in the formulas for populations with randon mating and (0.5 -R) in selfed populations, occupy corresponding positions. a) Heterotic Genes. (Quandro 9 and Fig. 7). RA menor do que 0,5 Ra menor do que 0,5 At equilibrium both genes remain in the populations and the three zygotic genotypes reach standard values, indepenr dent from the initial frequencies. b) In all other cases, i. e. where one or both values of the survival indices are equal or bigger than 0,5, the following situation arises: When the survival values for both ho-mozygotes are equal, equilibrium will be reached when the population contains only these homozygous with identical. When the survival values are different, equilibrium will be reached when only the homozygotes of the genotype whith the higher survival value remain in the population, while the other homozygotes and all heterozygotes have disappeared. 9) The result of variations in the initialgene frequencies (Fig. 8), of sudden changes in survival values (Fig. 9) and of the mode of reproduction (Fig. 10) are discussed in some detail. 10) References to problems of applied genetics. After some general observations on population genetics and applied genetics, two problems were discussed in some detail: a) Homogenization: - Formulas and data (Quadros 11 e 12) are given which show that random mating systeme are very inefficient means to obtain homozigous populations, and that either sslfing or some system of consanguineous matings should always be applied. These tables may also serve to obtain estimates as to the number of inbred generations necessary to obtain certain levels of homozygosis. b) Heterosis may evidently be obtained by the classical method of "inbreeding and outbreeding" or by the establishement of balanced populations containing genes with a heterotic effect, both with regards to their general vigor and productivity, as to their survival values. 11) Considerations on the evolutionary mechanismes. a)The importance of heterotic survival values is discussed, and references are made to a special publication on the subject (Brieger, 1948). b) Recessive lethals and semilethals. It is evident from the discussions that genes of this nature should be eliminated from the populations. But observations in corn, and in Drosophila, leave little doubt that genes such as tassel-seed, barren-stalk, numerous types of defective seeds, etc., are relatively frequent in populations which have not been subjected previously to a scientific selection. In order to explain this discrepancy, new studies are necessary in oder to find out whether survival in homozygotes, are not at the same time heterotic for the survival values. c) Dominant semilethals - It must be expected that these genes are speedily eliminated from the populations, and in fact, they are rare in natural populations. b) Incompatible genes - These genes also should be very rare, except perhaps in the special case where they serve to establish an isolating mechanism.Very Utile is known howewer with respect of them.