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This paper deals with problems on population genetics in Hymenoptera and particularly in social Apidae. 1) The studies on populations of Hymenoptera were made according to the two basic types of reproduction: endogamy and panmixia. The populations of social Apinae have a mixed method of reproduction with higher percentage of panmixia and a lower of endogamy. This is shown by the following a) males can enter any hive in swarming time; b) males of Meliponini are expelled from hives which does not need them, and thus, are forced to look for some other place; c) Meliponini males were seen powdering themselves with pollen, thus becoming more acceptable in any other hive. The panmixia is not complete owing to the fact that the density of the breeding population as very low, even in the more frequent species as low as about 2 females and 160 males per reproductive area. We adopted as selection values (or survival indices) the expressions according to Brieger (1948,1950) which may be summarised as follows; a population: p2AA + ²pq Aa + q2aa became after selection: x p2AA + 2pq Aa + z q²aa. For alge-braics facilities Brieger divided the three selective values by y giving thus: x/y p2 AA + y/y 2 pq Aa + z/y q²aa. He called x/y of RA and z/y of Ra, that are survival or selective index, calculated in relation to the heterozygote. In our case all index were calculated in relation to the heterozygote, including the ones for haploid males; thus we have: RA surveval index of genotype AA Ra surveval index of genotype aa R'A surveval index of genotype A R'a surveval index of genotype a 1 surveval index of genotype Aa The index R'A ande R'a were equalized to RA and Ra, respectively, for facilities in the conclusions. 2) Panmitic populations of Hymenoptera, barring mutations, migrations and selection, should follow the Hardy-Weinberg law, thus all gens will be present in the population in the inicial frequency (see Graphifc 1). 3) Heterotic genes: If mutation for heterotic gene ( 1 > RA > Ra) occurs, an equilibrium will be reached in a population when: P = R A + Ra - 2R²a _____________ (9) 2(R A + Ra - R²A - R²a q = R A + Ra - 2R²A _____________ (10) 2(R A + Ra - R²A - R²a A heterotic gene in an hymenopteran population may be maintained without the aid of new mutation only if the survival index of the most viable mutant (RA) does not exced the limiting value given by the formula: R A = 1 + √1+Ra _________ 4 If RA has a value higher thah the one permitted by the formula, then only the more viable gene will remain present in the population (see Graphic 10). The only direct proof for heterotic genes in Hymenoptera was given by Mackensen and Roberts, who obtained offspring from Apis mellefera L. queens fertilized by their own sons. Such inbreeding resulted in a rapid loss of vigor the colony; inbred lines intercrossed gave a high hybrid vigor. Other fats correlated with the "heterosis" problem are; a) In a colony M. quadrifasciata Lep., which suffered severely from heat, the percentage of deths omong males was greater .than among females; b) Casteel and Phillips had shown that in their samples (Apis melifera L). the males had 7 times more abnormalities tian the workers (see Quadros IV to VIII); c) just after emerging the males have great variation, but the older ones show a variation equal to that of workers; d) The tongue lenght of males of Apis mellifera L., of Bombus rubicundus Smith (Quadro X), of Melipona marginata Lep. (Quadro XI), and of Melipona quadrifasciata Lep. Quadro IX, show greater variationthan that of workers of the respective species. If such variation were only caused by subviables genes a rapid increasse of homozigoty for the most viable alleles should be expected; then, these .wild populations, supposed to be in equilibrium, could .not show such variability among males. Thus we conclude that heterotic genes have a grat importance in these cases. 4) By means of mathematical models, we came to the conclusion tht isolating genes (Ra ^ Ra > 1), even in the case of mutations with more adaptability, have only the opor-tunity of survival when the population number is very low (thus the frequency of the gene in the breeding population will be large just after its appearence). A pair of such alleles can only remain present in a population when in border regions of two races or subspecies. For more details see Graphics 5 to 8. 5) Sex-limited genes affecting only females, are of great importance toHymenoptera, being subject to the same limits and formulas as diploid panmitic populations (see formulas 12 and 13). The following examples of these genes were given: a) caste-determining genes in the genus Melipona; b) genes permiting an easy response of females to differences in feeding in almost all social Hymenoptera; c) two genes, found in wild populations, one in Trigona (Plebéia) mosquito F. SMITH (quadro XII) and other in Melipona marginata marginata LEP. (Quadro XIII, colonies 76 and 56) showing sex-limited effects. Sex-limited genes affecting only males do not contribute to the plasticity or genie reserve in hymenopteran populations (see formula 14). 6) The factor time (life span) in Hymenoptera has a particular importance for heterotic genes. Supposing one year to be the time unit and a pair of heterotic genes with respective survival indice equal to RA = 0, 90 and Ra = 0,70 to be present; then if the life time of a population is either one or two years, only the more viable gene will remain present (see formula 11). If the species has a life time of three years, then both alleles will be maintained. Thus we conclude that in specis with long lif-time, the heterotic genes have more importance, and should be found more easily. 7) The colonies of social Hymenoptera behave as units in competition, thus in the studies of populations one must determine the survival index, of these units which may be subdivided in indice for egg-laying, for adaptive value of the queen, for working capacity of workers, etc. 8) A study of endogamic hymenopteran populations, reproduced by sister x brother mating (fig. 2), lead us to the following conclusions: a) without selection, a population, heterozygous for one pair of alleles, will consist after some generations (theoretically after an infinite number of generation) of females AA fecundated with males A and females aa fecundated with males a (see Quadro I). b) Even in endogamic population there is the theoretical possibility of the presence of heterotic genes, at equilibrium without the aid of new mutations (see Graphics 11 and 12), but the following! conditions must be satisfied: I - surveval index of both homozygotes (RA e Ra) should be below 0,75 (see Graphic 13); II - The most viable allele must riot exced the less viable one by more than is permited by the following formula (Pimentel Gomes 1950) (see Gra-fic 14) : 4 R5A + 8 Ra R4A - 4 Ra R³A (Ra - 1) R²A - - R²a (4 R²a + 4 Ra - 1) R A + 2 R³a < o Considering these two conditions, the existance of heterotic genes in endogamic populations of Hymenoptera \>ecames very improbable though not - impossible. 9) Genie mutation offects more hymenopteran than diploid populations. Thus we have for lethal genes in diploid populations: u = q2, and in Hymenoptera: u = s, being u the mutation ratio and s the frequency of the mutant in the male population. 10) Three factors, important to competition among species of Meliponini were analysed: flying capacity of workers, food gathering capacity of workers, egg-laying of the queen. In this connection we refer to the variability of the tongue lenght observed in colonies from several localites, to the method of transporting the pollen in the stomach, from some pots (Melliponi-ni storage alveolus) to others (e. g. in cases of pillage), and to the observation that the species with the most populous hives are almost always the most frequent ones also. 11) Several defensive ways used for Meliponini to avoid predation are cited, but special references are made upon the camouflage of both hive (fig. 5) and hive entrance (fig. 4) and on the mimetism (see list in page ). Also under the same heading we described the method of Lestrimelitta for pillage. 12) As mechanisms important for promoting genetic plasticity of hymenopteran species we cited: a) cytological variations and b) genie reserve. As to the former, duplications and numerical variations of chromosomes were studied. Diprion simile ATC was cited as example for polyploidy. Apis mellife-ra L. (n = 16) also sugests polyploid origen since: a) The genus Melipona, which belongs to a" related tribe, presents in all species so far studied n = 9 chromosomes and b) there occurs formation of dyads in the firt spermatocyte division. It is su-gested that the origin of the sex-chromosome of Apis mellifera It. may be related to the possible origin of diplo-tetraploidy in this species. With regards to the genie reserve, several possible types of mutants were discussed. They were classified according to their survival indices; the heterotic and neutral mutants must be considered as more important for the genie reserve. 13) The mean radius from a mother to a daghter colony was estimated as 100 meters. Since the Meliponini hives swarm only once a year we may take 100 meters a year as the average dispersion of female Meliponini in ocordance to data obtained from Trigona (tetragonisca) jaty F. SMITH and Melipona marginata LEP., while other species may give different values. For males the flying distance was roughly estimated to be 10 times that for females. A review of the bibliography on Meliponini swarm was made (pg. 43 to 47) and new facts added. The population desity (breeding population) corresponds in may species of Meliponini to one male and one female per 10.000 square meters. Apparently the males are more frequent than the females, because there are sometimes many thousands, of males in a swarm; but for the genie frequency the individuals which have descendants are the ones computed. In the case of Apini and Meliponini, only one queen per hive and the males represented by. the spermatozoos in its spermateca are computed. In Meliponini only one male mate with the queen, while queens of Apis mellijera L. are fecundated by an average of about 1, 5 males. (Roberts, 1944). From the date cited, one clearly sees that, on the whole, populations of wild social bees (Meliponini) are so small that the Sewall Wright effect may become of great importance. In fact applying the Wright's formula: f = ( 1/aN♂ + 1/aN♀) (1 - 1/aN♂ + 1/aN♀) which measures the fixation and loss of genes per generation, we see that the fixation or loss of genes is of about 7% in the more frequent species, and rarer species about 11%. The variation in size, tergite color, background color, etc, of Melipona marginata Lep. is atributed to this genetic drift. A detail, important to the survival of Meliponini species, is the Constance of their breeding population. This Constance is due to the social organization, i. e., to the care given to the reproductive individuals (the queen with its sperm pack), to the way of swarming, to the food storage intended to control variations of feeding supply, etc. 14) Some species of the Meliponini are adapted to various ecological conditions and inhabit large geographical areas (e. g. T. (Tetragonisca jaty F. SMITH), and Trigona (Nanno-trigona testaceicornis LEP.) while others are limited to narrow regions with special ecological conditions (e. g. M. fuscata me-lanoventer SCHWARZ). Other species still, within the same geographical region, profit different ecological conditions, as do M. marginata LEP. and M. quadrifasciata LEP. The geographical distribution of Melipona quadrifasciata LEP. is different according to the subspecies: a) subsp anthidio-des LEP. (represented in Fig. 7 by black squares) inhabits a region fron the North of the S. Paulo State to Northeastern Brazil, ,b) subspecies quadrifasciata LEP., (marked in Fig. 7 with black triangles) accurs from the South of S. Paulo State to the middle of the State of Rio Grande do Sul (South Brazil). In the margined region between these two areas of distribution, hi-brid colonies were found (Fig. 7, white circles); they are shown with more details in fig. 8, while the zone of hybridization is roughly indicated in fig. 9 (gray zone). The subspecies quadrifasciata LEP., has 4 complete yellow bands on the abdominal tergites while anthidioides LEP. has interrupted ones. This character is determined by one or two genes and gives different adaptative properties to the subspecies. Figs. 10 shows certains meteorological isoclines which have aproximately the same configuration as the limits of the hybrid zone, suggesting different climatic adaptabilities for both genotypes. The exis-tance of a border zone between the areas of both subspecies, where were found a high frequency of hybrids, is explained as follows: being each subspecies adapted to a special climatic zone, we may suppose a poor adaptation of either one in the border region, which is also a region of intermediate climatic conditions. Thus, the hybrids, having a combination of the parent qualities, will be best adapted to the transition zone. Thus, the hybrids will become heterotic and an equilibrium will be reached with all genotypes present in the population in the border region.
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