On the adapted equations for several dyploid model in population genetics
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For a locus with two alleles ($I^A$ and $I^B$), the frequencies of the alleles are represented by \[ p=f(I^A)= \frac {2N_{AA} +N_{AB}} {2N} ,~~q=f(I^B)= \frac {2N_{BB} +N_{AB}}{2N} \] where $N_{AA},~N_{AB}$ and $N_{BB}$ are the numbers of $I^A I^A ,~I^A I^B$ and $I^B I^B$ respectively and $N$ is the total number of populations. The frequencies of the genotypes expected are calculated by using $p^2 ,~2pq$ and $q^2$. Choi showed the method of whether some genotypes is in these probabalities. Also he calculate the probability generating function for offspring number of genotype under a diploid model(\cite {Choi}). In this paper, let $x(t,p)$ be the probability that $I^A$ become fixed in the population by time $t$-th generation, given that its initial frequency at time $t=0$ is $p$. We find adapted equations for $x$ using the mean change of frequence of alleles and fitness of genotype. Also we apply this adapted equations to several diploid model and it also will apply to actual examples.
Abstract
For a locus with two alleles ($I^A$ and $I^B$), the frequencies of the alleles are represented by \[ p=f(I^A)= \frac {2N_{AA} +N_{AB}} {2N} ,~~q=f(I^B)= \frac {2N_{BB} +N_{AB}}{2N} \] where $N_{AA},~N_{AB}$ and $N_{BB}$ are the numbers of $I^A I^A ,~I^A I^B$ and $I^B I^B$ respectively and $N$ is the total number of populations. The frequencies of the genotypes expected are calculated by using $p^2 ,~2pq$ and $q^2$. Choi showed the method of whether some genotypes is in these probabalities. Also he calculate the probability generating function for offspring number of genotype under a diploid model(\cite {Choi}). In this paper, let $x(t,p)$ be the probability that $I^A$ become fixed in the population by time $t$-th generation, given that its initial frequency at time $t=0$ is $p$. We find adapted equations for $x$ using the mean change of frequence of alleles and fitness of genotype. Also we apply this adapted equations to several diploid model and it also will apply to actual examples.
참고문헌 (6)
- 최원, "On the genotype frequencies and generating function for frequencies in a dyploid model", 한국수학논문집, 29(1), pp.75~80 (2021), 강원경기수학회
- S.N.Ethier, "A class of degenerate diffusion processes occurring in population genetics", Communications on Pure and Applied Mathematics, XXIX (1976)
- J. H. Gillespie, "Natural selection for within-generation variance in offspring number", Genetics, 76, pp.601~606 (1974)
- M.Kimura, "A Stochastic model of Compensatory Neutral Evolution", Proceedings of a Workshop held in Nagoya (1985), Stochastic Methods in Biology
- R. Lewis, "Human Genetics : Concepts and Applications" (2016), McGraw-Hill Education
- B.A. Pierce, "Genetics Essentials : Concepts and Connections", pp.216~240 (2014), W.H.Freeman and Company