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--- chapter_19 [2024/09/09 23:25] – [Example of the effect of selection on recessive mutations: PKU] mike
+++ chapter_19 [2024/09/18 17:52] (current) – [Cystic fibrosis] mike
@@ Line 1: / Line 1: @@
-<typo fs:x-large>Chapter 19. Mutation, allele frequency, and selection</typo>
+<- chapter_18|Chapter 18^table_of_contents|Table of Contents^chapter_20|Chapter 20 ->
+<typo fs:x-large>%%Chapter 19. Mutation, allele frequency, and selection%%</typo>
 In the [[chapter_18|Chapter 18]] we saw that in a population, allele frequency does not change from generation to generation, unless:
@@ Line 33: / Line 35: @@
 </div>
-If a genotype has S = 0.75, then fitness = 0.25, meaning that individuals with this genotype will reproduce at a rate of only 25% relative to an average individual. Fitness can be thought of as a combination of survival and fertility. Let's put this into the context of the Hardy-Weinberg equilibrium. Recall from [[chapter_18|Chap. 18]] that $f(\frac{A}{A})=p^2$, $f(\frac{A}{a})=2pq$, and $f(\frac{a}{a})=q^2$.
+If a genotype has S = 0.75, then fitness = 0.25, meaning that individuals with this genotype will reproduce at a rate of only 25% relative to an average individual. Fitness can be thought of as a combination of survival and fertility. Let's put this into the context of the Hardy-Weinberg equilibrium. Recall from [[chapter_18|Chap. 18]] that $f(A/A)=p^2$, $f(A/a)=2pq$, and $f(a/a)=q^2$.
 <table Tab1>
 <columns 100% *100%*>
 ^  genotype  ^  frequency  ^  after selection  ^  change in frequency (Δ)  ^
-|  $\frac{A}{A}$  |  $p^2$  |  $p^2$  |  0  |
+|  $A/A$  |  $p^2$  |  $p^2$  |  0  |
-|  $\frac{A}{a}$  |  $2pq$  |  $2pq$  |  0  |
+|  $A/a$  |  $2pq$  |  $2pq$  |  0  |
-|  $\frac{a}{a}$  |  $q^2$  |  $q^2(1-s)$  |  $q^2(1-S)-q^2 = -Sq^2$  |
+|  $a/a$  |  $q^2$  |  $q^2(1-S)$  |  $q^2(1-S)-q^2 = -Sq^2$  |
 </columns>
 <caption>
@@ Line 56: / Line 58: @@
 </figure>
-For PKU, $q^2 = 10^{-4}$, so $q=10^{-2}$. Also, since PKU is fairly severe, during the course of human evolution pre-modern medicine $S \approx 1$. Therefore, based on Fig. {{ref>Fig2}} the estimated value of μ is about 10<sup>-4</sup>. The actual mutation frequency is probably not this high – and the relatively high $q$ for PKU is probably due to a founder effect in the European population or due to a balanced polymorphism (see below).
+For PKU, $q^2 = 10^{-4}$, so $q=10^{-2}$. Also, since PKU is fairly severe, in the pre-modern medicine age of human evolution $S \approx 1$ (that is, just about everyone who had PKU died before they could reproduce). Therefore, based on Fig. {{ref>Fig2}} the estimated value of μ is about 10<sup>-4</sup>. The actual mutation frequency is probably not this high – and the relatively high $q$ for PKU is probably due to a founder effect in the European population or due to a balanced polymorphism (see below).
-In modern times PKU can be treated by a low-phenylalanine diet so $S < 1$. This suggests that the frequency of PKU mutant alleles should start to rise at a rate $\mu = 10^{-4}$. Thus, $q$ will only increase by about a factor of 1% per generation. It will take a long time for this change in environment to have a significant effect on disease frequency.
+In modern times PKU can be treated by a low-phenylalanine diet; this means that in modern times $S << 1$ (or, you could say that $S \approx 0$). In this case, $\Delta q_{sel} = -Sq^2 \approx 0$ as well, and the main thing that will alter allele frequency would be the mutation rate μ. This suggests that the frequency of PKU mutant alleles should start to rise at a rate of $\mu = 10^{-4}$ per generation. Thus, $q$ will only increase by about a factor of 0.01% per generation. It will take a long time for this change in environment to have a significant effect on disease frequency.
-===== Example of the effect of selection on dominant mutations =====
+===== Example of the effect of selection on dominant mutations: Huntington's disease =====
-Now let’s determine the steady state allele frequency for a dominant disease with allele frequency $q = f(A)$. In contrast to the situation for recessive alleles, selection will operate against heterozygotes for dominant alleles. For rare dominant traits, almost all affected individuals area heterozygotes; that is, $f(\frac{A}{A})$ is very small. Therefore, while formally $q=f(\frac{A}{A})+\frac{1}{2}f\frac{A}{a})$, we can approximate $q$ by saying that $q \approx  \frac{1}{2}f\frac{A}{a})$. Let's look at how $S$ and $(1-S)$ affect $q$:
+Now let’s determine the steady state allele frequency for a dominant disease with allele frequency $q = f(A)$. In contrast to the situation for recessive alleles, selection will operate against heterozygotes for dominant alleles. For rare dominant traits, almost all affected individuals area heterozygotes; that is, $f(A/A)$ is very small. Therefore, while formally $q=f(A/A)+\frac{1}{2}f(A/a)$, we can approximate $q$ by saying that $q \approx  \frac{1}{2}f(A/a)$. Let's look at how $S$ and $(1-S)$ affect $q$:
 <table Tab2>
 <columns 100% *100%*>
 ^  genotype  ^  frequency  ^  after selection  ^  change in frequency (Δ)  ^
-|  $\frac{A}{A}$  |  -  |  -  |  -  |
+|  $A/A$  |  -  |  -  |  -  |
-|  $\frac{A}{a}$  |  $2pq \approx 2q$  |  $(1-S)2q$  |  $(1-S)2q-2q=-2Sq$  |
+|  $A/a$  |  $2pq \approx 2q$  |  $(1-S)2q$  |  $(1-S)2q-2q=-2Sq$  |
-|  $\frac{a}{a}$  |  $p^2$  |  $p^2$  |  0  |
+|  $a/a$  |  $p^2$  |  $p^2$  |  0  |
 </columns>
 <caption>
@@ Line 79: / Line 81: @@
 <figure>
-$$\begin{aligned}\Delta q_{sel}=\frac{1}{2}\Delta f(\frac{A}{a})&=\frac{1}{2}(-2Sq)\\&=-Sq\end{aligned}$$
+$$\begin{aligned}\Delta q_{sel}=\frac{1}{2}\Delta f(A/a)&=\frac{1}{2}(-2Sq)\\&=-Sq\end{aligned}$$
 <caption>
 placeholder
@@ Line 101: / Line 103: @@
 When $S<1$, the frequency of mutant alleles $q$ can get quite high (this makes sense mathematically; look at Fig. xx). A good example of this is Huntington's disease, which is caused by dominant mutations in $Htn$, the gene that codes for the Huntingtin protein. This devastating disease causes late onset neuromuscular degeneration starting at around 36 years of age, eventually leading to death. This obviously is terrible for anyone that is unfortunate enough to be a carrier, but since the disease doesn't manifest until later in life it doesn't decrease reproductive fitness much.
-===== Example of the effect of selection on sex-linked mutations =====
+===== Example of the effect of selection on sex-linked mutations: hemophilia A and DMD =====
 For the final example of a balance between mutation and selection, consider an X-linked recessive disease allele with frequency $q = f(a)$. For rare alleles the vast majority of affected individuals who are operated on by selection are males, and new mutations will increase the allele frequency (i.e., $\Delta q_{mut} \approx \mu$).
@@ Line 134: / Line 136: @@
 ===== Balanced polymorphisms =====
-Finally, we will consider a situation in which an allele is deleterious in the homozygous state but is beneficial in the heterozygous state. The steady state value of μ will be set by a balance between selection for the heterozygote and selection against the homozygote. We will need a new parameter, $h$ (the heterozygote advantage), that represents the increased reproductive fitness of heterozygote over an average individual.
+Finally, we will consider a situation in which an allele is deleterious in the homozygous state but is beneficial in the heterozygous state. The steady state value of $q$ will be set by a balance between selection for the heterozygote and selection against the homozygote. We will need a new parameter, $h$ (the heterozygote advantage), that represents the increased reproductive fitness of heterozygote over an average individual.
 <table>
 <columns 100% *100%*>
 ^  genotype  ^  frequency  ^  after selection  ^  change in frequency (Δ)  ^
-|  $\frac{A}{A}$  |  $p^2$  |  $p^2$  |  0  |
+|  $A/A$  |  $p^2$  |  $p^2$  |  0  |
-|  $\frac{A}{a}$  |  $2pq \approx 2q$  |  $(1+h)2q$  |  $(1+h)2q-2q=2hq$  |
+|  $A/a$  |  $2pq \approx 2q$  |  $(1+h)2q$  |  $(1+h)2q-2q=2hq$  |
-|  $\frac{a}{a}$  |  $q^2$  |  $(1-S)q^2$  |  $(1-S)q^2-q^2=-Sq^2$  |
+|  $a/a$  |  $q^2$  |  $(1-S)q^2$  |  $(1-S)q^2-q^2=-Sq^2$  |
 </columns>
 <caption>
-placeholder
+placeholder: $h$ is given as $0 \leq h \leq 1$; therefore, percent increase is given as $(1+h)$.
 </caption>
 </table>
@@ Line 150: / Line 152: @@
 As before when considering dominant mutations, when we calculate $\Delta q_{sel}$ we have to halve the change in frequency for heterozygotes, since only half the alleles of heterozygotes are mutant.
-$$\begin{aligned}\Delta q_{sel} &=\Delta f(\frac{a}{a})+\frac{1}{2}\Delta f(\frac{A}{a})\\
+$$\begin{aligned}\Delta q_{sel} &=\Delta f(a/a)+\frac{1}{2}\Delta f(A/a)\\
 &=-Sq^2+\frac{1}{2}(2hq)\\
 &=-Sq^2+hq \end{aligned}$$
@@ Line 166: / Line 168: @@
 ==== Cystic fibrosis ====
-A second example of balanced polymorphism is cystic fibrosis, a disease caused by autosomal recessive mutations in the $CFTR$ gene (__c__ystic __f__ibrosis __t__ransmembrane conductance __r__egulator). Mutants disrupt Cl<sup>–</sup> transport, leading to disturbed osmotic balance across in epithelial cell layers of the lungs and intestine. The incidence in European populations is approx. 0.0025; therefore, $q=\sqrt{0.0025}=0.05$. This is a pretty high frequency! This is probably not due to either high mutation frequency or founder effect (many different $CTFR$ disease alleles have been found although 70% are the ΔF508 allele). Scientists believe that heterozygotes may be more resistant to bacterial infections that cause diarrhea such as typhoid or cholera and that this selection was imposed in densely populated European cities.
+A second example of balanced polymorphism is cystic fibrosis, a disease caused by autosomal recessive mutations in the $CFTR$ gene (__c__ystic __f__ibrosis __t__ransmembrane conductance __r__egulator). Mutations in $CTFR$ disrupt Cl<sup>–</sup> transport, leading to disturbed osmotic balance across in epithelial cell layers of the lungs and intestine. The incidence in European populations is approx. 0.0025; therefore, $q=\sqrt{0.0025}=0.05$. This is a pretty high frequency! This is probably not due to either high mutation frequency or founder effect (many different $CTFR$ disease alleles have been found although 70% are the ΔF508 allele). Scientists believe that heterozygotes may be more resistant to bacterial infections that cause diarrhea such as typhoid or cholera and that this selection was imposed in densely populated European cities.
 ==== Lysosomal storage disorders ====