alleleS = c('A1', 'A2')
ntoss = 10 #the number of offspring produced
random_allele = sample(alleleS, size = ntoss, replace = TRUE)
table(random_allele)random_allele
A1 A2
5 5 Week7 E115L: Genetic Drift
by Grace Yuh Chwen Lee, Spring 2024
Allele frequencies could change due to selection for or against an allele, as what we learned last week. But even in the absence of selection, allele frequencies could still change due to randomness, known as genetic drift. There are difference sources of genetic drift, including
An individual carrying a particular allele can have more or less offspring than the average
Mendelian transmission
Source of genetic drift
Let’s look at what we worked on last week (diploid case) for the first source of randomness.
| A11 | A12 | A22 | |
|---|---|---|---|
| rate of survival | 100% | 100% | 80% |
| number of offspring | 50 | 70 | 50 |
| absolute fitness | ? | ? | ? |
| relative fitness | ? | ? | ? |
For an individual with A11 genotype, how many offspring can that individual produce?
How about another individual with A11 genotype?
Do you think individuals with A11 genotype ALWAYS produce exactly that number of offspring?
How about for other genotypes?
Let’s look at another source of randomness - Mendelian transmission. For heterozygous A12 individuals…
What is the probability that the A1 allele is passed down? How about A2 allele?
What if an A12 individual only produces one offspring, which allele will be passed down?
We can use the coin tossing example in week 4 (but now, selecting A1 or A2 allele) to illustrate this point.
Let’s write it into a function to make it easier to use.
per_A1_next_gen = function(n_offspring){
alleleS = c('A1', 'A2')
random_allele = sample(alleleS, size = n_offspring, replace = TRUE)
n_A1 = sum(random_allele == 'A1')
return(n_A1/n_offspring)
}
#only one offspring
per_A1_next_gen(1)[1] 1#5 offspring
per_A1_next_gen(5)[1] 0.8#ten offspring
per_A1_next_gen(10)[1] 0.7#a thousand offspring
per_A1_next_gen(1000)[1] 0.512Because of these two sources of randomness, even in the absence of selection/fitness differences, allele frequencies change from generations to generations. In this class, we will write simulations to explore how various factors influence how strong genetic drift changes allele frequencies.
Binomial distribution
Before we write our own simualtion, we will need to first learn about random variables with binomial distribution. A binomial random variable is the number of success in an experiment of N trails.
Let’s take coin tossing as an example (again). The “number of heads with N toss” has a distribution of binomial distribution.
coin_head = function(ntoss){
coin = c("head", "tail")
random_coin = sample(coin, size = ntoss, replace = TRUE)
no_head = sum(random_coin == "head")
return(no_head)
}
v_no_head = c()
ntoss = 100
for(i in 1:10000){
no_head = coin_head(ntoss)
v_no_head[i] = no_head
}
hist(v_no_head, xlab = "number of heads (100 toss)", main = "number of heads, fair coin")
Let’s consider a case of “unfair” coin, and compare the distributions.
coin_head = function(ntoss){
coin = c("head", "tail")
random_coin = sample(coin, size = ntoss, replace = TRUE, prob = c(0.9, 0.1))
no_head = sum(random_coin == "head")
return(no_head)
}
v_no_head2 = c()
ntoss = 100
for(i in 1:10000){
no_head = coin_head(ntoss)
v_no_head2[i] = no_head
}
hist(v_no_head2, xlab = "number of heads (100 toss)", main = "number of heads, unfair coin")
The two alleles, A1 and A2, could be like the “head” and “tail” of a unfair coin, with their probability being sampled following their frequencies.
To generate a random variable with a binomial distribution, we use the rbinom() function.
#the number of individuals with A1 allele, which has allele frequency of 0.9
#the number of individuals in the population is 100
#only do this sampling once
rbinom(n = 1, size = 100, prob = 0.9)[1] 91In class exercise - a simulator for genetic drift (this will be part of your lab report)
Key characteristic of genetic drift is “randomness”
Let’s try executing your drift_fun() 10 times, save it in a matrix, and plot all the results out in the same figure.
p = 0.5
N = 100
ngen = 200
m_p = matrix(nrow = 10, ncol = ngen)
for(i in 1:10){
v_p = drift_fun(p, N, ngen)
m_p[i,] = v_p
}
plot(m_p[1,], type = 'l', ylim = c(0, 1), ylab = "A1 frequency", xlab = "generation")
for(i in 2:10){
points(m_p[i,], type = 'l')
}
With the same
p,N, andngen, do you think the trajectories are similar between the 10 runs?Do you think the frequencies are more similar or more different as time goes by?
Impacts of genetic drift over generations
Let’s use some statistics to formally investigate whether the allele frequencies are indeed more different over time. We will learn how to use apply function in R.
?apply
v_mean = apply(m_p, 2, mean)
v_var = apply(m_p, 2, var)
#mean A1 frequency over time
plot(v_mean, pch = 20, xlab = "generation", ylab = "mean A1 freq.", ylim = c(0, 1), main = "mean A1 freq")
lines(c(0, ngen), c(0.5, 0.5), col = "red")
#var A1 frequency over time
plot(v_var, pch = 20, xlab = "generation", ylab = "var A1 freq.", main = "variance A1 freq")
From the two plots, do you think mean A1 frequency changes much over time?
How about variance of A1 frequency and what does that mean?
Let’s run the simulation even longer to see if we could observe even more dramatic effects.
p = 0.5
N = 100
ngen = 2000
m_p = matrix(nrow = 10, ncol = ngen)
for(i in 1:10){
v_p = drift_fun(p, N, ngen)
m_p[i,] = v_p
}
v_mean = apply(m_p, 2, mean)
v_var = apply(m_p, 2, var)
#mean A1 frequency over time
plot(v_mean, pch = 20, xlab = "generation", ylab = "mean A1 freq.", ylim = c(0, 1), main = "mean A1 freq")
lines(c(0, ngen), c(0.5, 0.5), col = "red")
#var A1 frequency over time
plot(v_var, pch = 20, xlab = "generation", ylab = "var A1 freq.", main = "mean A1 freq")
Why variance does not change anymore? Let’s plot out trajectories of individual run.
plot(m_p[1,], type = 'l', ylim = c(0, 1), ylab = "A1 frequency", xlab = "generation")
for(i in 2:10){
points(m_p[i,], type = 'l')
}
A1 frequencies either goes to fixation (i.e., p = 1, and A2 is lost) or lost (i.e., p = 0, and A1 is lost) after 500 generations.
Do you think genetic drift enhance or erode genetic diversity?
Impacts of initial allele frequency (p)
Let’s define another function, where it would return nrun number of series of drift simulations as a matrix.
multi_drift_fun = function(p, N, ngen, nrun){
multi_m_p = matrix(nrow = nrun, ncol = ngen)
for(i in 1:nrun){
v_p = drift_fun(p, N, ngen)
multi_m_p[i,] = v_p
}
return(multi_m_p)
}
N = 100
ngen = 200
nrun = 10 #number of simulations
#trying a series of initial allele frequencies
#p = 0.5 (the same as above), 0.1, 0.8
m_p_0.5 = multi_drift_fun(0.5, N, ngen, nrun)
m_p_0.1 = multi_drift_fun(0.1, N, ngen, nrun)
m_p_0.8 = multi_drift_fun(0.8, N, ngen, nrun)
#p = 0.5
plot(m_p_0.5[1,], type = 'l', ylim = c(0, 1), ylab = "A1 frequency", xlab = "generation", main = "p0 = 0.5")
for(i in 2:10){
points(m_p_0.5[i,], type = 'l')
}
#p = 0.1
plot(m_p_0.1[1,], type = 'l', ylim = c(0, 1), ylab = "A1 frequency", xlab = "generation", main = "p0 = 0.1")
for(i in 2:10){
points(m_p_0.1[i,], type = 'l')
}
#p = 0.8
plot(m_p_0.8[1,], type = 'l', ylim = c(0, 1), ylab = "A1 frequency", xlab = "generation", main = "p0 = 0.8")
for(i in 2:10){
points(m_p_0.8[i,], type = 'l')
}
Comparing these three plots with different starting allele frequencies of A1 (p), which one most likely leads to the lost of A1 allele?
Which one most likely leads to the fixation of A1 allele (i.e., lost of A2)?