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An introduction to the mathematical structure of the Wright-Fisher model of population genetics.

Tran TD, Hofrichter J, Jost J - Theory Biosci. (2012)

Bottom Line: In this paper, we develop the mathematical structure of the Wright-Fisher model for evolution of the relative frequencies of two alleles at a diploid locus under random genetic drift in a population of fixed size in its simplest form, that is, without mutation or selection.We establish a new concept of a global solution for the diffusion approximation (Fokker-Planck equation), prove its existence and uniqueness and then show how one can easily derive all the essential properties of this random genetic drift process from our solution.Thus, our solution turns out to be superior to the local solution constructed by Kimura.

View Article: PubMed Central - PubMed

Affiliation: Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany. trandat@mis.mpg.de

ABSTRACT
In this paper, we develop the mathematical structure of the Wright-Fisher model for evolution of the relative frequencies of two alleles at a diploid locus under random genetic drift in a population of fixed size in its simplest form, that is, without mutation or selection. We establish a new concept of a global solution for the diffusion approximation (Fokker-Planck equation), prove its existence and uniqueness and then show how one can easily derive all the essential properties of this random genetic drift process from our solution. Thus, our solution turns out to be superior to the local solution constructed by Kimura.

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Behaviour of the new solution from δp to pδ1 + (1 − p)δ0 in time with p = 0.4
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Fig1: Behaviour of the new solution from δp to pδ1 + (1 − p)δ0 in time with p = 0.4

Mentions: This new solution continuously deforms the initial state δp(x) (the allele A1 has relative frequency p) to pδ1(x) + (1 − p)δ0(x) (allele A1 is fixed with probability p and A2 is fixed with probability 1 − p) as time proceeds from 0 to In fact, the sequence {um(x,t)}m ≥ 0 satisfying25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \begin{aligned} u_m(x,t)&=\sum\limits_{i=0}^m c_i X_i(x){\hbox{e}}^{-\lambda_i t}\\ &\quad +\left\{1-p+\sum\limits_{i=0}^m c_i \left(-\frac{1}{2}\right){\hbox{e}}^{-\lambda_i t}\right\} \frac{m}{\sqrt{2\pi}}{\hbox{e}}^{-x^2 m^2/2}\\ &\quad +\left\{p+\sum\limits_{i= 0}^m c_i \left(\frac{(-1)^{i+1}}{2}\right){\hbox{e}}^{-\lambda_i t}\right\} \frac{m}{\sqrt{2 \pi}}{\hbox{e}}^{-(1-x)^2 m^2/2} \end{aligned} $$\end{document}tends to u for Therefore, we can visualise the asymptotic behaviour with the help of Mathematica (Fig. 1).Fig. 1


An introduction to the mathematical structure of the Wright-Fisher model of population genetics.

Tran TD, Hofrichter J, Jost J - Theory Biosci. (2012)

Behaviour of the new solution from δp to pδ1 + (1 − p)δ0 in time with p = 0.4
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4269093&req=5

Fig1: Behaviour of the new solution from δp to pδ1 + (1 − p)δ0 in time with p = 0.4
Mentions: This new solution continuously deforms the initial state δp(x) (the allele A1 has relative frequency p) to pδ1(x) + (1 − p)δ0(x) (allele A1 is fixed with probability p and A2 is fixed with probability 1 − p) as time proceeds from 0 to In fact, the sequence {um(x,t)}m ≥ 0 satisfying25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \begin{aligned} u_m(x,t)&=\sum\limits_{i=0}^m c_i X_i(x){\hbox{e}}^{-\lambda_i t}\\ &\quad +\left\{1-p+\sum\limits_{i=0}^m c_i \left(-\frac{1}{2}\right){\hbox{e}}^{-\lambda_i t}\right\} \frac{m}{\sqrt{2\pi}}{\hbox{e}}^{-x^2 m^2/2}\\ &\quad +\left\{p+\sum\limits_{i= 0}^m c_i \left(\frac{(-1)^{i+1}}{2}\right){\hbox{e}}^{-\lambda_i t}\right\} \frac{m}{\sqrt{2 \pi}}{\hbox{e}}^{-(1-x)^2 m^2/2} \end{aligned} $$\end{document}tends to u for Therefore, we can visualise the asymptotic behaviour with the help of Mathematica (Fig. 1).Fig. 1

Bottom Line: In this paper, we develop the mathematical structure of the Wright-Fisher model for evolution of the relative frequencies of two alleles at a diploid locus under random genetic drift in a population of fixed size in its simplest form, that is, without mutation or selection.We establish a new concept of a global solution for the diffusion approximation (Fokker-Planck equation), prove its existence and uniqueness and then show how one can easily derive all the essential properties of this random genetic drift process from our solution.Thus, our solution turns out to be superior to the local solution constructed by Kimura.

View Article: PubMed Central - PubMed

Affiliation: Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany. trandat@mis.mpg.de

ABSTRACT
In this paper, we develop the mathematical structure of the Wright-Fisher model for evolution of the relative frequencies of two alleles at a diploid locus under random genetic drift in a population of fixed size in its simplest form, that is, without mutation or selection. We establish a new concept of a global solution for the diffusion approximation (Fokker-Planck equation), prove its existence and uniqueness and then show how one can easily derive all the essential properties of this random genetic drift process from our solution. Thus, our solution turns out to be superior to the local solution constructed by Kimura.

Show MeSH