Fitness consequences of the selfish supergene Segregation Distorter

Function to make the mating table

The “mating table” is list of all the possible mating types (e.g. wild-type mother with a SD-heterozygote father, etc etc), which lists the possible offspring genotypes and their associated frequencies. The precise values in the mating table are a function of \(k\) (i.e. the strength of segregation distortion) and the sex ratio bias in heterozygote males. We assume that the meiosis is Mendelian except in SD-heterozygote males, and that all cross except those with a SD-heterozygote father produce a 50:50 sex ratio.

make_mating_table <- function(k, mating_types, SR_bias_in_hetero_males){

  offspring <- vector(mode = "list", nrow(mating_types))
  
  make_offspring <- function(row, type, zygote_freq){
    data.frame(mother = mating_types$mother[row],
               father = mating_types$father[row],
               type, 
               zygote_freq, 
               stringsAsFactors = FALSE)
  }
  
  g0 <- c("wild_type_female", "wild_type_male",
          "hetero_female_maternal", "hetero_male_maternal")
  g1 <- c("wild_type_female", "wild_type_male",
          "hetero_female_paternal", "hetero_male_paternal")
  g2 <- c("wild_type_female", "wild_type_male",
          "hetero_female_maternal", "hetero_male_maternal",
          "hetero_female_paternal", "hetero_male_paternal",
          "homo_female", "homo_male")
  g3 <- c("hetero_female_maternal", "hetero_male_maternal",
          "homo_female", "homo_male")
  g4 <- c("hetero_female_maternal", "hetero_male_maternal",
          "homo_female", "homo_male")
  p1 <- c(0.25*(1 - k)*(1 + SR_bias_in_hetero_males), 0.25*(1 - k)*(1 - SR_bias_in_hetero_males), 
          0.25*(1 + k)*(1 + SR_bias_in_hetero_males), 0.25*(1 + k)*(1 - SR_bias_in_hetero_males))
  p2 <- c(0.125*(1 - k)*(1 + SR_bias_in_hetero_males), 0.125*(1 - k)*(1 - SR_bias_in_hetero_males),
          0.125*(1 - k)*(1 + SR_bias_in_hetero_males), 0.125*(1 - k)*(1 - SR_bias_in_hetero_males),
          0.125*(1 + k)*(1 + SR_bias_in_hetero_males), 0.125*(1 + k)*(1 - SR_bias_in_hetero_males),
          0.125*(1 + k)*(1 + SR_bias_in_hetero_males), 0.125*(1 + k)*(1 - SR_bias_in_hetero_males))
  
  rep25 <- rep(0.25, 4)
  rep5  <- rep(0.5, 2)
  
  offspring[[1]]  <- make_offspring(1, c("wild_type_female", "wild_type_male"), rep5)
  offspring[[2]]  <- make_offspring(2, g0, rep25)
  offspring[[3]]  <- make_offspring(3, g0, rep25)
  offspring[[4]]  <- make_offspring(4, c("hetero_female_maternal", "hetero_male_maternal"), rep5)
  offspring[[5]]  <- make_offspring(5, g1, p1)
  offspring[[6]]  <- make_offspring(6, g2, p2)
  offspring[[7]]  <- make_offspring(7, g2, p2)
  offspring[[8]]  <- make_offspring(8, g3, p1)
  offspring[[9]]  <- make_offspring(9, g1, p1)
  offspring[[10]] <- make_offspring(10, g2, p2)
  offspring[[11]] <- make_offspring(11, g2, p2)
  offspring[[12]] <- make_offspring(12, g3, p1)
  offspring[[13]] <- make_offspring(13, c("hetero_female_paternal", "hetero_male_paternal"), rep5)
  offspring[[14]] <- make_offspring(14, g4, rep25)
  offspring[[15]] <- make_offspring(15, g4, rep25)
  offspring[[16]] <- make_offspring(16, c("homo_female", "homo_male"), rep5)
  
  output <-  do.call("rbind", offspring)
  
  names(offspring) <- paste(mating_types[,1], mating_types[,2], sep = " x ")
  output
}

mating_types <- expand.grid(mother = c("wild_type_female",
                                       "hetero_female_maternal",
                                       "hetero_female_paternal",
                                       "homo_female"),
                            father = c("wild_type_male",
                                       "hetero_male_maternal",
                                       "hetero_male_paternal",
                                       "homo_male"),
                            stringsAsFactors = FALSE) 
make_mating_table(
  k = 0.9, 
  mating_types, 
  SR_bias_in_hetero_males = 0.4) %>%
  kable() %>%
  kable_styling() %>%
  scroll_box(height = "500px")

Function to determine the mating type frequencies for the population

Given a set of genotype frequencies, it is simple to calculate the frequencies of each mating type. First, we implement selection, such that each genotype is represented in the mating types according to the product of its frequency and its fitness. Second, we determine the frequency of cross between male genotype \(i\) and female genotype \(j\) as \(i\times j\).

find_mating_type_frequencies <- function(pop, mating_types){
  
  # Implement selection
  pop$prop <- pop$prop * pop$fitness
  pop$prop <- pop$prop / sum(pop$prop)
  
  # Mating type freq is f_1 * f_2, post selection
  mating_types %>%
    mutate(mating_freq = pop$prop[match(mating_types$mother, pop$type)] * pop$prop[match(mating_types$father, pop$type)])
}

Define the main simulation function

This function iterates over generations, implementing selection and reproduction each time, until either A) the SD allele fixes, B) the SD allele goes extinct, or C) the generation timer expires.

# Helper function to calculate the frequency of the SD allele
calc_prop_SD <- function(pop){
  (sum(pop$prop[1:4]) + 2 * sum(pop$prop[5:6])) / 2
}

run_simulation <- function(generations,
                           k,
                           w_hetero_female_maternal,
                           w_hetero_female_paternal,
                           w_hetero_male_maternal,
                           w_hetero_male_paternal,
                           w_homo_male,
                           w_homo_female,
                           SR_bias_in_hetero_males,
                           initial_freq_SD,
                           mating_types){
  
  # Make the initial population
  pop <- data.frame(
    type = c("wild_type_female",
             "wild_type_male",
             "hetero_female_maternal",
             "hetero_male_maternal",
             "hetero_female_paternal",
             "hetero_male_paternal",
             "homo_female",
             "homo_male"),
    prop = c(rep(0.5  * (1 - initial_freq_SD)^2, 2),
             rep(0.25 * (1 - initial_freq_SD) * initial_freq_SD, 2),
             rep(0.25 * (1 - initial_freq_SD) * initial_freq_SD, 2),
             rep(0.5  * initial_freq_SD^2, 2)), 
    fitness = c(1, 1, w_hetero_female_maternal,
                w_hetero_female_paternal,
                w_hetero_male_maternal,
                w_hetero_male_paternal,
                w_homo_male,
                w_homo_female),
    stringsAsFactors = FALSE) %>% 
    arrange(type) 
  
  # Make the mating table used for this simulation run
  mating_table <- make_mating_table(k, mating_types, SR_bias_in_hetero_males)
  
  # Iterate over generations
  for(i in 1:generations){
    
    # Find the mating type frequencies, find the offspring frequencies, and renormalise the frequencies to sum to one
    prop_col <- mating_table %>%
      left_join(find_mating_type_frequencies(pop, mating_types), 
                by = c("mother", "father")) %>%
      mutate(offspring_freq = mating_freq * zygote_freq) %>%
      group_by(type) %>%
      summarise(prop = sum(offspring_freq)) %>%
      mutate(prop = prop / sum(prop)) %>%
      pull(prop)
    
    # Calculate the frequency of SD. 
    # Quit early if SD fixed/extinct-ish
    prop_SD <- calc_prop_SD(pop %>% mutate(prop = prop_col))
    # if(is.na(prop_SD)) print(pop)   # FOR DEBUGGING
    pop <- pop %>% mutate(prop = prop_col)
        
    if(prop_SD > 0.99) return(pop) 
    if(prop_SD < 0.0001) return(pop)
  }
  pop
}

Define a wrapper used to run the main simulation on a data frame of parameters

A helper function that runs run_simulation() on a data frame of parameter values.

run_many_simulations <- function(parameters){
  
  mating_types <- expand.grid(mother = c("wild_type_female",
                                         "hetero_female_maternal",
                                         "hetero_female_paternal",
                                         "homo_female"),
                              father = c("wild_type_male",
                                         "hetero_male_maternal",
                                         "hetero_male_paternal",
                                         "homo_male"),
                              stringsAsFactors = FALSE) 
  
  # Carefully pass all the parameters to run_simulation()
  run_one_simulation <- function(row, parameters, mating_types){
    genotypes <- run_simulation(
      generations = parameters$generations[row],
      k = parameters$k[row],
      w_hetero_female_maternal = parameters$w_hetero_female_maternal[row],
      w_hetero_female_paternal = parameters$w_hetero_female_paternal[row],
      w_hetero_male_maternal = parameters$w_hetero_male_maternal[row],
      w_hetero_male_paternal = parameters$w_hetero_male_paternal[row],
      w_homo_male = parameters$w_homo_male[row],
      w_homo_female = parameters$w_homo_female[row],
      SR_bias_in_hetero_males = parameters$SR_bias_in_hetero_males[row],
      initial_freq_SD = parameters$initial_freq_SD[row],
      mating_types = mating_types
    )
    
    output <- data.frame(parameters[row, ]) %>% as_tibble()
    output$genotypes <- list(genotypes %>% select(-fitness))
    output
  }
  
  # Loop over all the different parameter spaces
  lapply(1:nrow(parameters), run_one_simulation, parameters = parameters, mating_types = mating_types) %>%
    do.call("rbind", .) %>% 
    mutate(prop_SD = map_dbl(genotypes, calc_prop_SD))
}

Define the parameter space to investigate

Used to define the parameter spaces that are plotted in Figure 3.

imprint_cost <- 0.2 
resolution <- 101
resolution_y <- 91
gen <- 10000

parameters <- rbind(
  expand.grid(
    generations = gen,
    k = seq(0, 1, length = resolution),
    w_hetero_female_maternal = 1,
    w_hetero_female_paternal = 1,
    w_hetero_male_maternal = 1,
    w_hetero_male_paternal = 1,
    w_homo_male = 1,
    w_homo_female = 1,
    facet = "Cost-free",
    SR_bias_in_hetero_males = seq(-0.9, 0.9, length = resolution_y),
    initial_freq_SD = 0.01
  ),
  expand.grid(
    generations = gen,
    k = seq(0, 1, length = resolution),
    w_hetero_female_maternal = .8,
    w_hetero_female_paternal = .8,
    w_hetero_male_maternal = .8,
    w_hetero_male_paternal = .8,
    w_homo_male = .8,
    w_homo_female = .8,
    facet = "Dominant cost",
    SR_bias_in_hetero_males = seq(-0.9, 0.9, length = resolution_y),
    initial_freq_SD = 0.01
  ),
  expand.grid(
    generations = gen,
    k = seq(0, 1, length = resolution),
    w_hetero_female_maternal = 1,
    w_hetero_female_paternal = 1,
    w_hetero_male_maternal = 1,
    w_hetero_male_paternal = 1,
    w_homo_male = 0,
    w_homo_female = 0,
    facet = "Recessive lethal",
    SR_bias_in_hetero_males = seq(-0.9, 0.9, length = resolution_y),
    initial_freq_SD = 0.01
  ),
  expand.grid(
    generations = gen,
    k = seq(0, 1, length = resolution),
    w_hetero_female_maternal = .8,
    w_hetero_female_paternal = .8,
    w_hetero_male_maternal = .8,
    w_hetero_male_paternal = .8,
    w_homo_male = 0,
    w_homo_female = 0,
    facet = "Recessive lethal plus heterozygote cost",
    SR_bias_in_hetero_males = seq(-0.9, 0.9, length = resolution_y),
    initial_freq_SD = 0.01
  ),
 expand.grid(
    generations = gen,
    k = seq(0, 1, length = resolution),
    w_hetero_female_maternal = 1,
    w_hetero_female_paternal = 1 - imprint_cost,
    w_hetero_male_maternal = 1,
    w_hetero_male_paternal = 1 - imprint_cost,
    w_homo_male = 1 - imprint_cost,
    w_homo_female = 1 - imprint_cost,
    facet = "SD is costly if paternally inherited (PI)",
    SR_bias_in_hetero_males = seq(-0.9, 0.9, length = resolution_y),
    initial_freq_SD = 0.01
  ),
  expand.grid(
    generations = gen,
    k = seq(0, 1, length = resolution),
    w_hetero_female_maternal = .8,
    w_hetero_female_paternal = .8 - imprint_cost,
    w_hetero_male_maternal = .8,
    w_hetero_male_paternal = .8 - imprint_cost,
    w_homo_male = .8 - imprint_cost,
    w_homo_female = .8 - imprint_cost,
    facet = "Dominant cost plus cost of PI",
    SR_bias_in_hetero_males = seq(-0.9, 0.9, length = resolution_y),
    initial_freq_SD = 0.01
  ),
  expand.grid(
    generations = gen,
    k = seq(0, 1, length = resolution),
    w_hetero_female_maternal = 1,
    w_hetero_female_paternal = 1 - imprint_cost,
    w_hetero_male_maternal = 1,
    w_hetero_male_paternal = 1 - imprint_cost,
    w_homo_male = 0,
    w_homo_female = 0,
    facet = "Recessive lethal plus cost of PI",
    SR_bias_in_hetero_males = seq(-0.9, 0.9, length = resolution_y),
    initial_freq_SD = 0.01
  ),
 expand.grid(
    generations = gen,
    k = seq(0, 1, length = resolution),
    w_hetero_female_maternal = .8,
    w_hetero_female_paternal = .8 - imprint_cost,
    w_hetero_male_maternal = .8,
    w_hetero_male_paternal = .8 - imprint_cost,
    w_homo_male = 0,
    w_homo_female = 0,
    facet = "Recessive lethal, heterozygote cost, and cost of PI",
    SR_bias_in_hetero_males = seq(-0.9, 0.9, length = resolution_y),
    initial_freq_SD = 0.01
  ),
  expand.grid(
    generations = gen,
    k = seq(0, 1, length = resolution),
    w_hetero_female_maternal = 1 - imprint_cost,
    w_hetero_female_paternal = 1,
    w_hetero_male_maternal = 1 - imprint_cost,
    w_hetero_male_paternal = 1,
    w_homo_male = 1 - imprint_cost,
    w_homo_female = 1 - imprint_cost,
    facet = "SD is costly if maternally inherited (MI)",
    SR_bias_in_hetero_males = seq(-0.9, 0.9, length = resolution_y),
    initial_freq_SD = 0.01
  ),
  expand.grid(
    generations = gen,
    k = seq(0, 1, length = resolution),
    w_hetero_female_maternal = .8 - imprint_cost,
    w_hetero_female_paternal = .8,
    w_hetero_male_maternal = .8 - imprint_cost,
    w_hetero_male_paternal = .8,
    w_homo_male = .8 - imprint_cost,
    w_homo_female = .8 - imprint_cost,
    facet = "Dominant cost plus cost of MI",
    SR_bias_in_hetero_males = seq(-0.9, 0.9, length = resolution_y),
    initial_freq_SD = 0.01
  ),
  expand.grid(
    generations = gen,
    k = seq(0, 1, length = resolution),
    w_hetero_female_maternal = 1 - imprint_cost,
    w_hetero_female_paternal = 1,
    w_hetero_male_maternal = 1 - imprint_cost,
    w_hetero_male_paternal = 1,
    w_homo_male = 0,
    w_homo_female = 0,
    facet = "Recessive lethal plus cost of MI",
    SR_bias_in_hetero_males = seq(-0.9, 0.9, length = resolution_y),
    initial_freq_SD = 0.01
  ),
  expand.grid(
    generations = gen,
    k = seq(0, 1, length = resolution),
    w_hetero_female_maternal = .8 - imprint_cost,
    w_hetero_female_paternal = .8,
    w_hetero_male_maternal = .8 - imprint_cost,
    w_hetero_male_paternal = .8,
    w_homo_male = 0,
    w_homo_female = 0,
    facet = "Recessive lethal, heterozygote cost, and cost of MI",
    SR_bias_in_hetero_males = seq(-0.9, 0.9, length = resolution_y),
    initial_freq_SD = 0.01
  )
) %>% as_tibble()

Run the simulation

if(file.exists("data/simulation_results.rds")){
  completed <- readRDS("data/simulation_results.rds") %>% select(-genotypes, -prop_SD) %>% select(!! names(parameters))
  parameters <- parameters[!(apply(parameters, 1, paste0, collapse = "") %in% apply(completed, 1, paste0, collapse = "")), ]
}

if(nrow(parameters) > 0){
  
  # If it's big, divide the job into 100 equal-ish chunks so that we can see the % completion as it proceeds
  cores <- 7
  if(nrow(parameters) > 10000)  split_parameters <- split(parameters, sample(1:100, nrow(parameters), replace = TRUE))
  else split_parameters <- list(parameters)
  
  for(i in 1:length(split_parameters)){
    print(i)
    plan("multicore")
    chunk_size <- floor(nrow(split_parameters[[i]]) / cores)
    simulation_output <- split(split_parameters[[i]], c(rep(1:(cores-1), each = chunk_size), 
                                                        rep(cores, nrow(split_parameters[[i]]) - chunk_size*(cores-1)))) %>%
      future_lapply(run_many_simulations) %>% do.call("rbind", .)
    saveRDS(simulation_output, paste("data/simulation_output_", sample(9999999999, 1), ".rds", sep = ""))
  }
  
 simulation_output <- list.files(path = "data", pattern = "simulation_output_", full.names = TRUE) %>%
   lapply(readRDS) %>% do.call("rbind", .)
 saveRDS(simulation_output, "data/simulation_results.rds")
  
} 
simulation_results <- readRDS("data/simulation_results.rds")
# delete individual files
# unlink(list.files(path = "data", pattern = "simulation_output_", full.names = TRUE))

Make Figure 3

fig_3 <- simulation_results %>% 
  mutate(percent_sons = 100 - 100 * (1 + SR_bias_in_hetero_males)/2) %>%
  ggplot(aes(k, percent_sons, fill = prop_SD)) +
  geom_blank() +
  geom_raster() + 
  stat_contour(aes(z = prop_SD), colour = "grey10") +
  geom_hline(yintercept = 50, colour = "black", linetype = 3) +
  labs(x = "Strength of segregation distortion (K)", 
       y = "Offspring sex ratio of male SD heterozygotes (% sons)") +
  facet_wrap(~ facet, labeller = labeller(facet = label_wrap_gen(35))) + 
  scale_fill_distiller(palette = "PuRd", direction =  1, name = "Equilibrium frequency of SD", limits = c(0,1)) +
  scale_x_continuous(expand = c(0, 0), labels = c(0, 0.25, 0.5, 0.75, 1)) + 
  scale_y_continuous(expand = c(0, 0), labels = ) +
  theme(panel.border = element_rect(fill = NA, colour = "black", size = .8),
        strip.background = element_rect(colour = "black", fill = "grey90", size = .8),
        legend.position = "bottom")

ggsave(fig_3, filename = "figures/fig3.pdf", height = 9, width = 9)
fig_3

Past versions of unnamed-chunk-9-1.png

Figure 3: The equilibrium frequency reached by the allele depends on the strength of segregataion distortion (\(K\) = 0 indicates fair meiosis, \(K\) = 1 means that heterozygotes transmit only the allele) and the direction and strength of sex ratio bias in the progeny of heterozygote males. The four columns make different assumptions about the fitness costs to individuals carrying the allele (see Results), while the three rows assume either that has no parent-of-origin-specific effects on fitness (top row), or that is especially costly when paternally inherited (middle row) or maternally inherited (bottom row).

Session information

sessionInfo()

R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.6

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRlapack.dylib

locale:
[1] en_AU.UTF-8/en_AU.UTF-8/en_AU.UTF-8/C/en_AU.UTF-8/en_AU.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] ggplot2_3.1.0      kableExtra_0.9.0   future.apply_1.0.1
[4] future_1.11.1.1    dplyr_0.8.0.1      purrr_0.3.2       

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.1         RColorBrewer_1.1-2 highr_0.8         
 [4] plyr_1.8.4         pillar_1.3.1.9000  compiler_3.5.1    
 [7] git2r_0.23.0       workflowr_1.3.0    tools_3.5.1       
[10] digest_0.6.18      gtable_0.2.0       viridisLite_0.3.0 
[13] evaluate_0.13      tibble_2.0.99.9000 pkgconfig_2.0.2   
[16] rlang_0.3.4        rstudioapi_0.10    yaml_2.2.0        
[19] parallel_3.5.1     xfun_0.6           withr_2.1.2       
[22] stringr_1.4.0      httr_1.4.0         knitr_1.22        
[25] xml2_1.2.0         fs_1.3.1           globals_0.12.4    
[28] hms_0.4.2          grid_3.5.1         rprojroot_1.3-2   
[31] tidyselect_0.2.5   glue_1.3.1.9000    listenv_0.7.0     
[34] R6_2.4.0           rmarkdown_1.10     readr_1.1.1       
[37] magrittr_1.5       whisker_0.3-2      scales_1.0.0      
[40] backports_1.1.2    codetools_0.2-15   htmltools_0.3.6   
[43] assertthat_0.2.1   rvest_0.3.2        colorspace_1.3-2  
[46] labeling_0.3       stringi_1.4.3      lazyeval_0.2.2    
[49] munsell_0.5.0      crayon_1.3.4