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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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Comparison results of expectation of the absorption time
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Fig4: Comparison results of expectation of the absorption time

Mentions: is the unique solution of the one-dimensional boundary value problem\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left\{\begin{array}{l} Lv=-1, \hbox { in (0,1)}\\ v(0)=v(1)=0. \end{array}\right. $$\end{document}We easily check that this agrees with our formula above by using Mathematica (Fig. 4):


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

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

Comparison results of expectation of the absorption time
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

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

Fig4: Comparison results of expectation of the absorption time
Mentions: is the unique solution of the one-dimensional boundary value problem\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left\{\begin{array}{l} Lv=-1, \hbox { in (0,1)}\\ v(0)=v(1)=0. \end{array}\right. $$\end{document}We easily check that this agrees with our formula above by using Mathematica (Fig. 4):

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