By Prof. Dr. Thomas Nagylaki (auth.)
Most of those notes have been offered as a part of a two-quarter direction on theoretical inhabitants genetics on the college of Chicago. just about all the scholars have been both undergraduates in arithmetic or graduate scholars within the organic sciences. the single must haves have been calculus and matrices. As is finished in those notes, organic historical past and extra mathematical suggestions have been lined once they have been required. i've got incorporated the suitable difficulties assigned within the path. My objective in those notes is to formulate many of the versions really typically, making the organic assumptions relatively specific, and to accomplish the analyses rather conscientiously. i am hoping the alternative and remedy of issues will allow the reader to appreciate and evaluation distinct analyses of particular versions and functions within the literature. No test has been made to check the literature or to assign credits. lots of the references are to papers at once germane to the topics and ways lined the following. Frequency of reference isn't meant to mirror proportionate contribution. i'm very thankful to Professor James F. Crow for priceless reviews and to Mrs. Adelaide Jaffe for her very good typing. I thank the nationwide technological know-how starting place for its help (Grant No. DEB76-01550).
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Additional info for Selection in One- and Two-Locus Systems
Ii) pet) 0 + We can have either Case 2 or Case 3 above. It is necessary to separate the case of recessive A. 38) PW 12 /W 22 , which yields the geometric rate = (const) t (w 12 /w 22 ) ' t + 00. If for weak selection (k « 1) the rate is roughly (const)e -kt . 59 To examine the elimination of a recessive, we must approximate . 38) to second order in p. The failure of the linear approximation leading to a geometric rate is not surprising: selection is only against AA, and the frequency of these individuals, p2, is extremely small for p «1.
Then the adults 1. J 1. J will be in Hardy-Weinberg proportions. 1, the gene frequencies in zygotes in the next generation would be Pi' and the Hardy-Weinberg zygotic frequencies would be PiPj' This is the standard viability selection model. Our more general formulation will demonstrate that the form of the equation'for gene frequency change is always the same, and Hardy-Weinberg proportions apply with multiplicative fertilities. 6), we see that the zygotic proportions in the next generation are r -1 P!.
The equilibrium P is stable, but not asymptotically stable. Before ending this section, we shall briefly consider mutation. Let u i j be the probability per unit time that A. mutates to A. for '!. J i F j. 2, u i i = O. 80) uij · To include selection, we let selection act continuously~ and 30 suppose mutation occurs every ~t time units. 80) we obtain p~(t+~t) 1. p. (tHt) 1. (t) 1. 1. -m)~t + O[(~t)2], 1. (tHt) + 1. (t+~t)u •• J J1. ~ UijJ~t + O[(~t)2]. 82) with two alleles for the problems. In most organisms, it is necessary to distinguish zygotic and germ line genotypic frequencies, and to take into account that selection acts on the phenotype, which develops from the offspring genotype, and germ cells mutate without phenotypic effect (Nagylaki, 1974).
Selection in One- and Two-Locus Systems by Prof. Dr. Thomas Nagylaki (auth.)