Week 5 assignment 4

1. Vectorising functions

To complete this part of the assignment, you need to define the calc_p() function from the tutorial:

calc_p <- function(counts){
  # get the number of samples
  n <- sum(counts)
  # calculate frequency of 1st allele - p
  p <- ((counts[1]*2) + counts[2])/(2*n)
  return(p)
}

You have the following data frame containing genotype counts from four populations:

counts <- data.frame(
  A1A1 = c(10, 0, 84, 32),
  A1A2 = c(30, 48, 13, 15),
  A2A2 = c(75, 60, 3, 28),
  location = c("loc1", "loc2", "loc3", "loc4")
)

  1. Use calc_p() together with apply() to calculate \(p\) for each population. Add the values to counts as a column. hint: you have to select only the numeric columns of the data frame to use apply() on it.

  2. Use ifelse() to create a column that says “above 0.5” if p in the population is larger than 0.5, and “below 0.5” if it’s not.

You should print the data frame in the end and show the output in the hand-in.

2. A worked example of \(F_{ST}\)

We sample two fish populations - one in the lake and the other in a stream. We genotype them at locus B. In the lake, the genotype counts are - B1B1 = 32, B1B2 = 12 and B2B2 = 6. In the stream, the genotype counts are B1B1 =10, B1B2 = 16, B2B2 = 43.

  1. Calculate the average expected heterozygosity for the two populations.

    1. Optional: make a function to calculate average expected heterozygosity.

  2. Calculate the expected heterozygosity for the two populations as a metapopulation.

    1. Optional: make a function to calculate expected heterozygosity in the metapopulation.

  3. Calculate \(F_{st}\) between the lake and stream fish. How do you interpret this value?

3. More on \(F_{ST}\)

To complete this part of the assignment, you need to define the calc_fst() function, in addition to the calc_p() function defined in question 1.

calc_fst <- function(p_1, p_2){
  
  # calculate q1 and q2
  q_1 <- 1 - p_1
  q_2 <- 1 - p_2
  
  # calculate total allele frequency
  p_t <- (p_1 + p_2)/2
  q_t <- 1 - p_t
  
  # calculate expected heterozygosity
  # first calculate expected heterozygosity for the two populations
  # pop1
  hs_1 <- 2*p_1*q_1
  # pop2
  hs_2 <- 2*p_2*q_2
  # then take the mean of this
  hs <- (hs_1 + hs_2)/2
  
  # next calculate expected heterozygosity for the metapopulations
  ht <- 2*p_t*q_t
  
  # calculate fst
  fst <- (ht - hs)/ht
  
  # return output
  return(fst)
}

  1. Using the calc_p() and calc_fst() functions we developed during the tutorial and the lct_freq data, calculate \(F_{ST}\) between the Han_China and the Swedish_and_Finnish_Scandinavia populations. What might be a biological explanation of the \(F_{ST}\) value you calculate? Hint: think about what the LCT gene does, and the geographical patterns of lactose intolerance.

  2. Using the functions we developed in the tutorial, calculate \(F_{ST}\) around the LCT gene between Americans of European descent and also between African Americans. Plot this like we plotted the \(F_{ST}\) between European Americans and East Asians. What is the highest value of \(F_{ST}\)? How does this compare with the highest \(F_{ST}\) between European Americans and East Asians that we investigated in the tutorial?