5: Population Genetics
- Page ID
- 93486
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Deoxyribonucleic acid, or DNA-a large double-stranded, helical molecule, with rungs made from the four base pairs adenine (A), cytosine (C), thymine (T) and guanine \((G)\)-carries inherited genetic information. The ordering of the base pairs \(A, C\), \(\mathrm{T}\) and \(\mathrm{G}\) determines the DNA sequence. A gene is a particular DNA sequence that is the fundamental unit of heredity for a particular trait. Some species develop as diploids, carrying two copies of every gene, one from each parent, and some species develop as haploids with only one copy. There are even species that develop as both diploids and haploids.
Consider the pea plant, which develops as a diploid. When we say there is a gene for pea color, say, we mean there is a particular DNA sequence that may vary in a pea plant population, and that there are at least two subtypes, called alleles, where plants with two copies of the yellow-color allele have yellow peas, those with two copies of the green-color allele, green peas. A plant with two copies of the same allele is homozygous for that particular gene (or a homozygote), while a plant carrying two different alleles is heterozygous (or a heterozygote). For the pea color gene, a plant carrying both a yellow- and green-color allele has yellow peas. We say that the green color is a recessive trait (or the green-color allele is recessive), and the yellow color is a dominant trait (or the yellow-color allele is dominant). The combination of alleles carried by the plant is called its genotype, while the actual trait (green or yellow peas) is called its phenotype. A gene that has more than one allele in a population is called polymorphic, and we say the population has a polymorphism for that particular gene.
Population genetics can be defined as the mathematical modeling of the evolution and maintenance of polymorphism in a population. Population genetics together with Charles Darwin’s theory of evolution by natural selection and Gregor Mendel’s theory of biological inheritance forms the modern evolutionary synthesis (sometimes called the modern synthesis, the evolutionary synthesis, the neoDarwinian synthesis, or neo-Darwinism). The primary founders in the early twentieth century of population genetics were Sewall Wright, J. B. S. Haldane and Ronald Fisher.
Allele frequencies in a population can change due to the influence of four primary evolutionary forces: natural selection, genetic drift, mutation, and migration. Here, we mainly focus on natural selection and mutation. Genetic drift is the study of stochastic effects, and it is important in small populations. Migration typically requires consideration of the spatial distribution of a population, and it is usually modeled mathematically by partial differential equations.
The simplified models we will consider assume infinite population sizes (neglecting stochastic effects except in \(\$ 5.5\) ), well-mixed populations (neglecting any spatial distribution), and discrete generations (neglecting any age-structure). Our main purpose is to illustrate the fundamental ways that a genetic polymorphism can be maintained in a population.
| genotype | \(A\) | \(a\) |
| number | \(n_{A}\) | \(n_{a}\) |
| viability fitness | \(g_{A}\) | \(g_{a}\) |
| fertility fitness | \(f_{A}\) | \(f_{a}\) |