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We shall refer to segregation distorting alleles as SD and non-distorting wildtype alleles as W throughout this analysis. This notation can be found in our code chunks and in the accompanying explanations. For example, we refer to an individual homozygous for the SD allele as SDSD, a heterozygous individual as WSD and an individual homozygous for the wildtype allele as WW.

Load packages

library(tidyverse) # for purrr, ggplot, dplyr and more 
library(patchwork) # for combining plots
library(rcartocolor) # for cool colours
library(ggpubr) # for combining plots
library(kableExtra) # scrolling tables
library(pander) # other nice tables
library(stickylabeller) # for labelling ggplot facets
library(ggnewscale) # for multiple scales in a single plot

Enter empirical data from present study

$~$

In our empirical investigations, we have estimated the following parameters for the SD-5, SD-72 and SD-Mad alleles. We use these to parameterise our model, and to estimate the predicted equilibrium frequency for each SD allele, within the model’s parameter space (see Figure 2 and Figure S6)

k_c values.
$S_{precop}$, the mating success of SD/+ males (no measurements for SD-5).
$P1_{cost}$ and $P2_{cost}$ for WSD males*.
The probability of a female remating (albeit in very specific circumstances) after mating with a WSD male ($p_{WSD}$) or a WW male ($p_{WW}$).

*We did not measure the sperm competitive ability of SDSD homozygotes, because SD-5 and SD-72 are homozygous lethal. SD-Mad is homozygous viable, but a previous study (Wong and Holman, 2019) indicates that the fitness of SD-Mad homozygotes is very close to 0.

# k_c values

estimated_k_SD5 <- 0.944 # 94.4% of progeny inherit SD 

estimated_k_SD72 <- 0.909 # 90.9% of progeny inherit SD 

estimated_k_SDMad <- 0.868 # 86.8% of progeny inherit SD

# The relative mating success for each SD variant

estimated_precop_SD5 <- NA # not measured 

estimated_precop_SD72 <- 1.24
  
estimated_precop_SDMad <- 1.19

# P1cost and P2cost values, plugged in from Table S7b and Table S8b

P1_SD5 <- 1 - 0.15 

P2_SD5 <- 1 - 0.945 

P1_SD72 <- 1 - 0.43 

P2_SD72 <- 1 - 0.997 

P1_SDMad <- 1 - 0.39 

P2_SDMad <- 0 # no cost

# We also include 95% credible intervals for P1 and P2 cost, to use in Figure 2 and Figure S8

lower_P1_SD5 <- 1 - 0.02 

lower_P2_SD5 <- 1 - 0.855

lower_P1_SD72 <- 1 - 0.05 

lower_P2_SD72 <- 1 - 0.969 

lower_P1_SDMad <-  NA #1 - 0.04 essentially identical to SD-72 so to avoid clutter in our figures we don't include this value

lower_P2_SDMad <- 0 # no cost

upper_P1_SD5 <- 1 - 0.6

upper_P2_SD5 <- 1 - 0.997

upper_P1_SD72 <- 0 # no cost

upper_P2_SD72 <- 0 # no cost

upper_P1_SDMad <- 0 # no cost

upper_P2_SDMad <- 0 # no cost

# Remating probabilities from Table S9a, using estimates from Block 1

estimated_prob_remating_W = 0.304 

estimated_prob_remating_SD5 = 0.755 

estimated_prob_remating_SD72 = 0.425 

estimated_prob_remating_SDMad = 0.429

# Create a tibble containing empirical estimates

empirical_estimates <-
  tibble(SD = c("SD-5", "SD-72", "SD-Mad"),
         kc = c(estimated_k_SD5, estimated_k_SD72, estimated_k_SDMad),
         Precop = c(estimated_precop_SD5, estimated_precop_SD72, estimated_precop_SDMad),
         P1cost = c(P1_SD5, P1_SD72, P1_SDMad),
         lower_P1cost = c(lower_P1_SD5, lower_P1_SD72, lower_P1_SDMad),
         upper_P1cost = c(upper_P1_SD5, upper_P1_SD72, upper_P1_SDMad),
         P2cost = c(P2_SD5, P2_SD72, P2_SDMad),
         lower_P2cost = c(lower_P2_SD5, lower_P2_SD72, lower_P2_SDMad),
         upper_P2cost = c(upper_P2_SD5, upper_P2_SD72, upper_P2_SDMad),
         remating_prob = c(estimated_prob_remating_SD5, estimated_prob_remating_SD72, estimated_prob_remating_SDMad))

pander(empirical_estimates, split.cell = 40, split.table = Inf)

SD	kc	Precop	P1cost	lower_P1cost	upper_P1cost	P2cost	lower_P2cost	upper_P2cost	remating_prob
SD-5	0.944	NA	0.85	0.98	0.4	0.055	0.145	0.003	0.755
SD-72	0.909	1.24	0.57	0.95	0	0.003	0.031	0	0.425
SD-Mad	0.868	1.19	0.61	NA	0	0	0	0	0.429

$~$

Modelling segregation distortion and sperm competition costs

$~$

Here we describe how we found zygote frequencies when offspring were produced by individuals with different genotypes (WW, WSD or SDSD), considering both segregation distortion and sperm competition.

In our model we assume that SDSD individuals are homozygous lethal, as is the case with many SD variants (see modelling viability and precopulatory sexual selection). However, in the event that this assumption is relaxed, we assume that WW and SDSD homozygotes perform identically in sperm competition against themselves or each other. This is because it is assumed that the reduction in sperm number resulting from drive is the mechanism behind the loss in competitive ability, and drive is absent in viable SD homozygotes.

Parameter meanings:

$P1_{normal}$ is the average P1 for the population, where $0 \le P1_{normal} \le 1$.
$P1_{cost}$ is a multiplicative cost to the proportion of offspring sired when the SD male mates first, where $0 \le P1_{cost} \le 1$ (larger value = greater cost).
$P2_{cost}$ is a multiplicative cost to the proportion of offspring sired when the SD male mates second, where $0 \le P2_{cost} \le 1$ (larger value = greater cost).
For example, if we measured P1 for the SD/+ males as being 1/4 as high as for wildtype controls, we would set $P1_{cost}$ = 0.75 to recapitulate nature in the model
$k$ is the proportion of SD bearing sperm a female is inseminated with when mating with a SD/+ male.

$~$

Next we provide examples of how we calculated zygote frequencies for two of the potential mating combinations in our population. There are 36 possible mating combinations considered by our model and zygote frequencies for each of these combinations are displayed in the code chunk found under the heading Function to make the mating table (in the code these are named rep5, rep25 and p1-18).

$~$

1. Finding offspring genotypes when a WW female mates with a WW male then a WSD male

We think of the production of zygote genotypes as a two-step process. First, which male sires the offspring? Second, which of his possible sperm genotypes won? The second step can be ignored for WW males, but for the WSD males there is segregation distortion to consider. WW progeny can be created two ways - sired by the WW male, or sired by the WSD male, with one of his W sperm fertilising the egg. The WSD progeny can only be sired by the WSD male, and an SD sperm needs to fertilise the egg.

Scenario 1: W sperm from the WW male fertilises the W egg

A sperm from the WW male wins with probability: $P1_{normal} + (1 - P1_{normal}) * P2_{cost}$
This is the ‘normal’ or mean P1 value for the population, plus an additional amount due to an advantage in sperm competition, up to a maximum of 1 when $P2cost = 1$

Scenario 2: W sperm from the WSD male fertilises the W egg

A sperm from the WSD male wins with probability: $1 - (P1_{normal} + (1 - P1_{normal}) * P2_{cost})$ which simplifies to: $(P1_{normal} - 1) * (P2_{cost} - 1)$
A W sperm from this male wins with probability: $1 - k$
Multiply these probabilities to get the chance both occur, and you get: $(P1_{normal} - 1) * (P2_{cost} - 1) * (1 - k)$

Adding Scenario 1 and Scenario 2 gives the overall probability each offspring is WW:

$P1_{normal} + (1 - P1_{normal}) * P2_{cost} + (P1_{normal} - 1) * (P2_{cost} - 1) * (1 - k)$

Scenario 3: SD sperm from the WSD male fertilises the W egg, producing a zygote with the genotype WSD

$(P1_{normal} - 1) * (P2_{cost} - 1) * k$

$~$

2. Finding offspring genotypes when a WW female mates with a WSD male then a WW male (i.e. the reverse mating order)

Scenario 1: W sperm from the WW male fertilises the W egg

A sperm from the WW male wins with probability: $1 - (P1_{normal} - P1_{normal} * P1_{cost})$
Similar to Scenario 1 in the first mating trio example, this is P2 (or equivalently 1 - P1), plus an additional amount due to an advantage in sperm competition.

Scenario 2: W sperm from the WSD male fertilises the W egg

A sperm from the WSD male wins with probability: $P1_{normal} - P1_{normal} * P1_{cost}$
A W sperm from this male wins with probability: $1 - k$
Multiply these probabilities to get the chance both occur, and you get: $(P1_{normal} - P1_{normal} * P1_{cost}) * (1 - k)$

Adding Scenario 1 and Scenario 2 gives us the overall probability each offspring is WW:

$1 - (P1_{normal} - P1_{normal} * P1_{cost}) + (P1_{normal} - P1_{normal} * P1_{cost}) * (1 - k)$

Scenario 3: SD sperm from the WSD male wins, giving a WSD offspring

$(P1_{normal} - P1_{normal} * P1cost) * k$

$~$

Function to make the mating table

$~$

The “mating table” is list of all the possible mating types (e.g. a WSD mother with a WSD father, etc), the resulting 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 expected paternity of WSD males when the female is double-mated to a WW and WSD male, with specific paternity values depending on the order of mating (as described above). We assume that meiosis is Mendelian except in WSD males, and that all crosses produce a 50:50 sex ratio.

Explore the code chunk below to see how we calculated zygote frequencies for every possible mating combination (rep5, rep25 and p1-18).

make_mating_table <- function(k, normalP1, P1cost, P2cost){
  
  make_offspring <- function(X, Y, Z, type, zygote_freq){
    tibble(mother = X,
           father1 = Y,
           father2 = Z,
           type,
           zygote_freq)
  }
  
  # Set up the 36 possible mating trio types
  
  mating_types <- expand_grid(
    mother = c("WW_female",
               "WSD_female",
               "SDSD_female"),
    father1 = c("WW_male",
                "WSD_male",
                "SDSD_male"),
    father2 = c("none",
                "WW_male",
                "WSD_male",
                "SDSD_male"))
  
  # Specify the possible offspring genotypes for all the potential crosses; we use these for the type argument in the make_offspring function
  
  # WW x WW
  
  g0 <- c("WW_female", "WW_male")
  
  # WSD x WW & WW x WSD
  
  g1 <- c("WW_female", "WW_male", "WSD_female", "WSD_male")
  
  # SDSD x WW & WW x SDSD
  
  g2 <- c("WSD_female", "WSD_male")
  
  # WSD x WSD
  
  g3 <- c("WW_female", "WW_male", "WSD_female", "WSD_male", "SDSD_female", "SDSD_male")
  
  # WSD x SDSD & SDSD x WSD
  
  g4 <- c("WSD_female", "WSD_male", "SDSD_female", "SDSD_male")
  
  # SDSD x SDSD
  
  g5 <- c("SDSD_female", "SDSD_male")
  
  
  # Now calculate the zygote frequencies for each cross
  
  # even frequency of four offspring types
  
  rep25 <- rep(0.25, 4)
  
  # even frequency between two offspring types
  
  rep5 <- rep(0.5, 2)
  
  # now offspring frequencies when there's drive
  
  # when there are 4 offspring genotypes
  
  p1 <- c(0.5 * (1 - k), 
          0.5 * (1 - k), 
          0.5 * k, 
          0.5 * k)
  
  # when there are 6 offspring genotypes
  
  p2 <- c(0.25 * (1 - k),
          0.25 * (1 - k),
          (0.25 * (1 - k)) + (0.25 * k),
          (0.25 * (1 - k)) + (0.25 * k),
          0.25 * k,
          0.25 * k)
  
  # Now add sperm comp zygote frequencies
  
  # when WW mates with WW & WSD
  
  p3 <- c(0.5 * ((normalP1 + (1 - normalP1) * P2cost) + (normalP1 - 1) * (P2cost - 1) * (1-k)), # WW female offspring
          0.5 * ((normalP1 + (1 - normalP1) * P2cost) + (normalP1 - 1) * (P2cost - 1) * (1-k)), # WW male offspring
          0.5 * ((normalP1 - 1) * (P2cost - 1) * k), # WSD female offspring
          0.5 * ((normalP1 - 1) * (P2cost - 1) * k)) # WSD female offspring
  
  # when WW mates with WSD & WW
  
  p4 <- c(0.5 * ((normalP1 - normalP1 * P1cost) * (1 - k) + 1 - (normalP1 - normalP1 * P1cost)), # WW female offspring
          0.5 * ((normalP1 - normalP1 * P1cost) * (1 - k) + 1 - (normalP1 - normalP1 * P1cost)), # WW male offspring
          0.5 * ((normalP1 - normalP1 * P1cost) * k), # WSD females
          0.5 * ((normalP1 - normalP1 * P1cost) * k)) # WSD females
  
  
  # when WW mates with WW and SDSD & when SDSD mates with WW and SDSD
  
  p5 <- c(0.5 * normalP1, # WW females & WSD females
          0.5 * normalP1, # WW males & WSD males
          0.5 * (1 - normalP1), # WSD females and SDSD females
          0.5 * (1 - normalP1)) # WSD males and SDSD males
  
  # when WW mates with SDSD and WW & when SDSD mates with SDSD and WW
  
  p6 <- c(0.5 * (1 - normalP1), # WW females & WSD females
          0.5 * (1 - normalP1), # WW males & WSD males
          0.5 * normalP1, # WSD females & SDSD females
          0.5 * normalP1) # WSD males & SDSD females
  
  # when WSD mates with WW and WSD
  
  p7 <- c(0.5 * (((normalP1 + (1 - normalP1) * P2cost) * 0.5) + (normalP1 - 1) * (P2cost - 1) * (1 - k) * 0.5), # WW females
          0.5 * (((normalP1 + (1 - normalP1) * P2cost) * 0.5) + (normalP1 - 1) * (P2cost - 1) * (1 - k) * 0.5), # WW males
          0.5 * (((normalP1 + (1 - normalP1) * P2cost) * 0.5) + (normalP1 - 1) * (P2cost - 1) * (1 - k) * 0.5 + (normalP1 - 1) * (P2cost - 1) * k * 0.5), # WSD females
          0.5 * (((normalP1 + (1 - normalP1) * P2cost) * 0.5) + (normalP1 - 1) * (P2cost - 1) * (1 - k) * 0.5 + (normalP1 - 1) * (P2cost - 1) * k * 0.5), # WSD males
          0.5 * (normalP1 - 1) * (P2cost - 1) * k * 0.5, # SDSD females
          0.5 * (normalP1 - 1) * (P2cost - 1) * k * 0.5) # SDSD males
  
  
  # when WSD mates with WSD and WW
  
  p8 <- c(0.5 * ((normalP1 - normalP1 * P1cost) * (1 - k) * 0.5 + (1 - (normalP1 - normalP1 * P1cost)) * 0.5), # WW female 
          0.5 * ((normalP1 - normalP1 * P1cost) * (1 - k) * 0.5 + (1 - (normalP1 - normalP1 * P1cost)) * 0.5), # WW male
          0.5 * ((normalP1 - normalP1 * P1cost) * k * 0.5 + (normalP1 - normalP1 * P1cost) * (1 - k) * 0.5 + (1 - (normalP1 - normalP1 * P1cost)) * 0.5), # WSD female
          0.5 * ((normalP1 - normalP1 * P1cost) * k * 0.5 + (normalP1 - normalP1 * P1cost) * (1 - k) * 0.5 + (1 - (normalP1 - normalP1 * P1cost)) * 0.5), # WSD male
          0.5 * (normalP1 - normalP1 * P1cost) * k * 0.5, # SDSD female
          0.5 * (normalP1 - normalP1 * P1cost) * k * 0.5) # SDSD male
  
  
  # when WW mates with WSD and SDSD 
  # Because there is no drive we assume SDSD = WW for sperm comp
  
  p9 <- c( 0.5 * ((normalP1 - normalP1 * P1cost) * (1 - k)), # WW females
           0.5 * ((normalP1 - normalP1 * P1cost) * (1 - k)), # WW males
           0.5 * ((normalP1 - normalP1 * P1cost) * k + 1 - (normalP1 - normalP1 * P1cost)), # WSD female offspring
           0.5 * ((normalP1 - normalP1 * P1cost) * k + 1 - (normalP1 - normalP1 * P1cost))) # WSD male offspring
  
  # when WW mates with SDSD and WSD
  
  p10 <- c(0.5 * (normalP1 - 1) * (P2cost - 1) * (1 - k), # WW female offspring
           0.5 * (normalP1 - 1) * (P2cost - 1) * (1 - k), # WW male offspring
           0.5 * ((normalP1 + (1 - normalP1) * P2cost) + (normalP1 - 1) * (P2cost - 1) * k), # WSD female offspring
           0.5 * ((normalP1 + (1 - normalP1) * P2cost) + (normalP1 - 1) * (P2cost - 1) * k)) # WSD female offspring
  
  
  # when WSD mates with WW and SDSD
  
  p11 <- c(0.5 * (normalP1 * 0.5), # WW females
           0.5 * (normalP1 * 0.5), # WW males
           0.5 * ((normalP1 * 0.5) + (1 - normalP1) * 0.5), # WSD females
           0.5 * ((normalP1 * 0.5) + (1 - normalP1) * 0.5), # WSD males
           0.5 * ((1 - normalP1) * 0.5), # SDSD females
           0.5 * ((1 - normalP1) * 0.5)) # SDSD males
  
  
  # when WSD mates with SDSD and WW
  
  p12 <- c(0.5 * ((1 - normalP1) * 0.5), # WW females
           0.5 * ((1 - normalP1) * 0.5), # WW males
           0.5 * ((normalP1 * 0.5) + (1 - normalP1) * 0.5), # WSD females
           0.5 * ((normalP1 * 0.5) + (1 - normalP1) * 0.5), # WSD males
           0.5 * (normalP1 * 0.5), # SDSD females
           0.5 * (normalP1 * 0.5)) # SDSD males
  
  
  # WSD mates with WSD and SDSD
  
  p13 <- c(0.5 * (normalP1 - normalP1 * P1cost) * (1 - k) * 0.5, # WW female
           0.5 * (normalP1 - normalP1 * P1cost) * (1 - k) * 0.5, # WW male
           0.5 * ((normalP1 - normalP1 * P1cost) * k * 0.5 + (normalP1 - normalP1 * P1cost) * (1 - k) * 0.5 + (1 - (normalP1 - normalP1 * P1cost)) * 0.5), # WSD female
           0.5 * ((normalP1 - normalP1 * P1cost) * k * 0.5 + (normalP1 - normalP1 * P1cost) * (1 - k) * 0.5 + (1 - (normalP1 - normalP1 * P1cost)) * 0.5), # WSD male
           0.5 * ((normalP1 - normalP1 * P1cost) * k * 0.5 + (1 - (normalP1 - normalP1 * P1cost)) * 0.5), # SDSD female 
           0.5 * ((normalP1 - normalP1 * P1cost) * k * 0.5 + (1 - (normalP1 - normalP1 * P1cost)) * 0.5)) # SDSD male
  
  
  
  # WSD mates with SDSD and WSD
  
  p14 <- c(0.5 * (normalP1 - 1) * (P2cost - 1) * (1 - k) * 0.5, # WW females
           0.5 * (normalP1 - 1) * (P2cost - 1) * (1 - k) * 0.5, # WW males
           0.5 * (((normalP1 + (1 - normalP1) * P2cost) * 0.5) + (normalP1 - 1) * (P2cost - 1) * (1 - k) * 0.5 + (normalP1 - 1) * (P2cost - 1) * k * 0.5), # WSD females
           0.5 * (((normalP1 + (1 - normalP1) * P2cost) * 0.5) + (normalP1 - 1) * (P2cost - 1) * (1 - k) * 0.5 + (normalP1 - 1) * (P2cost - 1) * k * 0.5), # WSD males
           0.5 * (((normalP1 + (1 - normalP1) * P2cost) * 0.5) + (normalP1 - 1) * (P2cost - 1) * k * 0.5), # SDSD females
           0.5 * (((normalP1 + (1 - normalP1) * P2cost) * 0.5) + (normalP1 - 1) * (P2cost - 1) * k * 0.5)) # SDSD males
  
  
  # SDSD mates with WW and WSD
  
  p15 <- c(0.5 * ((normalP1 + (1 - normalP1) * P2cost) + (normalP1 - 1) * (P2cost - 1) * (1 - k)), # WSD females 
           0.5 * ((normalP1 + (1 - normalP1) * P2cost) + (normalP1 - 1) * (P2cost - 1) * (1 - k)), # WSD males 
           0.5 * (normalP1 - 1) * (P2cost - 1) * k, # SDSD females
           0.5 * (normalP1 - 1) * (P2cost - 1) * k) # SDSD males
  
  # SDSD mates with WSD and WW
  
  p16 <- c(0.5 * ((1 - (normalP1 - normalP1 * P1cost)) + (normalP1 - normalP1 * P1cost) * (1 - k)), # WSD females 
           0.5 * ((1 - (normalP1 - normalP1 * P1cost)) + (normalP1 - normalP1 * P1cost) * (1 - k)), # WSD males 
           0.5 * (normalP1 - normalP1 * P1cost) * k, # SDSD females
           0.5 * (normalP1 - normalP1 * P1cost) * k) # SDSD males
  
  
  # SDSD mates with WSD and SDSD
  
  p17 <- c(0.5 * (normalP1 - normalP1 * P1cost) * (1 - k), ## WSD female
           0.5 *(normalP1 - normalP1 * P1cost) * (1 - k), ## WSD male
           0.5 * ((normalP1 - normalP1 * P1cost) * k + (1 - (normalP1 - normalP1 * P1cost))), ## SDSD female
           0.5 * ((normalP1 - normalP1 * P1cost) * k + (1 - (normalP1 - normalP1 * P1cost)))) ## SDSD male
  
  
  # SDSD mates with SDSD and WSD
  
  p18 <- c(0.5 * ((normalP1 - 1) * (P2cost - 1) * (1 - k)), # WSD females
           0.5 * ((normalP1 - 1) * (P2cost - 1) * (1 - k)), # WSD males
           0.5 * ((normalP1 - 1) * (P2cost - 1) * k + (normalP1 + (1 - normalP1) * P2cost)), # SDSD females
           0.5 * ((normalP1 - 1) * (P2cost - 1) * k + (normalP1 + (1 - normalP1) * P2cost))) # SDSD males
  
  
  bind_rows(
    list(
      make_offspring("WW", "WW", "none", g0, rep5),
      make_offspring("WW", "WSD", "none", g1, p1),
      make_offspring("WW", "SDSD", "none", g2, rep5),
      make_offspring("WSD", "WW", "none", g1, rep25),
      make_offspring("WSD", "WSD", "none", g3, p2),
      make_offspring("WSD", "SDSD", "none", g4, rep25),
      make_offspring("SDSD", "WW", "none", g2, rep5),
      make_offspring("SDSD", "WSD", "none", g4, p1),
      make_offspring("SDSD", "SDSD", "none", g5, rep5),
      make_offspring("WW", "WW", "WW", g0, rep5),
      make_offspring("WW", "WW", "WSD", g1, p3), 
      make_offspring("WW", "WW", "SDSD", g1, p5),
      make_offspring("WW", "WSD", "WW", g1, p4),
      make_offspring("WW", "WSD", "WSD", g1, rep25),
      make_offspring("WW", "WSD", "SDSD", g1, p9),
      make_offspring("WW", "SDSD", "WW", g1, p6),
      make_offspring("WW", "SDSD", "WSD", g1, p10),
      make_offspring("WW", "SDSD", "SDSD", g2, rep5),
      make_offspring("WSD", "WW", "WW", g1, rep25),
      make_offspring("WSD", "WW", "WSD", g3, p7),
      make_offspring("WSD", "WW", "SDSD", g3, p11),
      make_offspring("WSD", "WSD", "WW", g3, p8),
      make_offspring("WSD", "WSD", "WSD", g3, p2),
      make_offspring("WSD", "WSD", "SDSD", g3, p13),
      make_offspring("WSD", "SDSD", "WW", g3, p12),
      make_offspring("WSD", "SDSD", "WSD", g3, p14),
      make_offspring("WSD", "SDSD", "SDSD", g4, rep25),
      make_offspring("SDSD", "WW", "WW", g2, rep5),
      make_offspring("SDSD", "WW", "WSD", g4, p15),
      make_offspring("SDSD", "WW", "SDSD", g4, p5),
      make_offspring("SDSD", "WSD", "WW", g4, p16),
      make_offspring("SDSD", "WSD", "WSD", g4, p1),
      make_offspring("SDSD", "WSD", "SDSD", g4, p17),
      make_offspring("SDSD", "SDSD", "WW", g4, p6),
      make_offspring("SDSD", "SDSD", "WSD", g4, p18),
      make_offspring("SDSD", "SDSD", "SDSD", g5, rep5)
    )) %>% 
    arrange(father2, mother, father1)
  
}

$~$

An example of a mating table

$~$

Assume $k$ = 0.9, and the following paternity values: $normal_{P1}$ = 0.1, $P1_{cost}$ = 0.5, and $P2_{cost}$ = 0.5

# Make the table

make_mating_table(k = 0.9, normalP1 = 0.1, P1cost = 0.5, P2cost = 0.5) %>% 
  rename(zygote_type = type) %>% 
  kable() %>% 
  kable_styling() %>% 
  scroll_box(height = "500px")

mother	father1	father2	zygote_type	zygote_freq
SDSD	SDSD	none	SDSD_female	0.50000
SDSD	SDSD	none	SDSD_male	0.50000
SDSD	WSD	none	WSD_female	0.05000
SDSD	WSD	none	WSD_male	0.05000
SDSD	WSD	none	SDSD_female	0.45000
SDSD	WSD	none	SDSD_male	0.45000
SDSD	WW	none	WSD_female	0.50000
SDSD	WW	none	WSD_male	0.50000
WSD	SDSD	none	WSD_female	0.25000
WSD	SDSD	none	WSD_male	0.25000
WSD	SDSD	none	SDSD_female	0.25000
WSD	SDSD	none	SDSD_male	0.25000
WSD	WSD	none	WW_female	0.02500
WSD	WSD	none	WW_male	0.02500
WSD	WSD	none	WSD_female	0.25000
WSD	WSD	none	WSD_male	0.25000
WSD	WSD	none	SDSD_female	0.22500
WSD	WSD	none	SDSD_male	0.22500
WSD	WW	none	WW_female	0.25000
WSD	WW	none	WW_male	0.25000
WSD	WW	none	WSD_female	0.25000
WSD	WW	none	WSD_male	0.25000
WW	SDSD	none	WSD_female	0.50000
WW	SDSD	none	WSD_male	0.50000
WW	WSD	none	WW_female	0.05000
WW	WSD	none	WW_male	0.05000
WW	WSD	none	WSD_female	0.45000
WW	WSD	none	WSD_male	0.45000
WW	WW	none	WW_female	0.50000
WW	WW	none	WW_male	0.50000
SDSD	SDSD	SDSD	SDSD_female	0.50000
SDSD	SDSD	SDSD	SDSD_male	0.50000
SDSD	WSD	SDSD	WSD_female	0.00250
SDSD	WSD	SDSD	WSD_male	0.00250
SDSD	WSD	SDSD	SDSD_female	0.49750
SDSD	WSD	SDSD	SDSD_male	0.49750
SDSD	WW	SDSD	WSD_female	0.05000
SDSD	WW	SDSD	WSD_male	0.05000
SDSD	WW	SDSD	SDSD_female	0.45000
SDSD	WW	SDSD	SDSD_male	0.45000
WSD	SDSD	SDSD	WSD_female	0.25000
WSD	SDSD	SDSD	WSD_male	0.25000
WSD	SDSD	SDSD	SDSD_female	0.25000
WSD	SDSD	SDSD	SDSD_male	0.25000
WSD	WSD	SDSD	WW_female	0.00125
WSD	WSD	SDSD	WW_male	0.00125
WSD	WSD	SDSD	WSD_female	0.25000
WSD	WSD	SDSD	WSD_male	0.25000
WSD	WSD	SDSD	SDSD_female	0.24875
WSD	WSD	SDSD	SDSD_male	0.24875
WSD	WW	SDSD	WW_female	0.02500
WSD	WW	SDSD	WW_male	0.02500
WSD	WW	SDSD	WSD_female	0.25000
WSD	WW	SDSD	WSD_male	0.25000
WSD	WW	SDSD	SDSD_female	0.22500
WSD	WW	SDSD	SDSD_male	0.22500
WW	SDSD	SDSD	WSD_female	0.50000
WW	SDSD	SDSD	WSD_male	0.50000
WW	WSD	SDSD	WW_female	0.00250
WW	WSD	SDSD	WW_male	0.00250
WW	WSD	SDSD	WSD_female	0.49750
WW	WSD	SDSD	WSD_male	0.49750
WW	WW	SDSD	WW_female	0.05000
WW	WW	SDSD	WW_male	0.05000
WW	WW	SDSD	WSD_female	0.45000
WW	WW	SDSD	WSD_male	0.45000
SDSD	SDSD	WSD	WSD_female	0.02250
SDSD	SDSD	WSD	WSD_male	0.02250
SDSD	SDSD	WSD	SDSD_female	0.47750
SDSD	SDSD	WSD	SDSD_male	0.47750
SDSD	WSD	WSD	WSD_female	0.05000
SDSD	WSD	WSD	WSD_male	0.05000
SDSD	WSD	WSD	SDSD_female	0.45000
SDSD	WSD	WSD	SDSD_male	0.45000
SDSD	WW	WSD	WSD_female	0.29750
SDSD	WW	WSD	WSD_male	0.29750
SDSD	WW	WSD	SDSD_female	0.20250
SDSD	WW	WSD	SDSD_male	0.20250
WSD	SDSD	WSD	WW_female	0.01125
WSD	SDSD	WSD	WW_male	0.01125
WSD	SDSD	WSD	WSD_female	0.25000
WSD	SDSD	WSD	WSD_male	0.25000
WSD	SDSD	WSD	SDSD_female	0.23875
WSD	SDSD	WSD	SDSD_male	0.23875
WSD	WSD	WSD	WW_female	0.02500
WSD	WSD	WSD	WW_male	0.02500
WSD	WSD	WSD	WSD_female	0.25000
WSD	WSD	WSD	WSD_male	0.25000
WSD	WSD	WSD	SDSD_female	0.22500
WSD	WSD	WSD	SDSD_male	0.22500
WSD	WW	WSD	WW_female	0.14875
WSD	WW	WSD	WW_male	0.14875
WSD	WW	WSD	WSD_female	0.25000
WSD	WW	WSD	WSD_male	0.25000
WSD	WW	WSD	SDSD_female	0.10125
WSD	WW	WSD	SDSD_male	0.10125
WW	SDSD	WSD	WW_female	0.02250
WW	SDSD	WSD	WW_male	0.02250
WW	SDSD	WSD	WSD_female	0.47750
WW	SDSD	WSD	WSD_male	0.47750
WW	WSD	WSD	WW_female	0.25000
WW	WSD	WSD	WW_male	0.25000
WW	WSD	WSD	WSD_female	0.25000
WW	WSD	WSD	WSD_male	0.25000
WW	WW	WSD	WW_female	0.29750
WW	WW	WSD	WW_male	0.29750
WW	WW	WSD	WSD_female	0.20250
WW	WW	WSD	WSD_male	0.20250
SDSD	SDSD	WW	WSD_female	0.45000
SDSD	SDSD	WW	WSD_male	0.45000
SDSD	SDSD	WW	SDSD_female	0.05000
SDSD	SDSD	WW	SDSD_male	0.05000
SDSD	WSD	WW	WSD_female	0.47750
SDSD	WSD	WW	WSD_male	0.47750
SDSD	WSD	WW	SDSD_female	0.02250
SDSD	WSD	WW	SDSD_male	0.02250
SDSD	WW	WW	WSD_female	0.50000
SDSD	WW	WW	WSD_male	0.50000
WSD	SDSD	WW	WW_female	0.22500
WSD	SDSD	WW	WW_male	0.22500
WSD	SDSD	WW	WSD_female	0.25000
WSD	SDSD	WW	WSD_male	0.25000
WSD	SDSD	WW	SDSD_female	0.02500
WSD	SDSD	WW	SDSD_male	0.02500
WSD	WSD	WW	WW_female	0.23875
WSD	WSD	WW	WW_male	0.23875
WSD	WSD	WW	WSD_female	0.25000
WSD	WSD	WW	WSD_male	0.25000
WSD	WSD	WW	SDSD_female	0.01125
WSD	WSD	WW	SDSD_male	0.01125
WSD	WW	WW	WW_female	0.25000
WSD	WW	WW	WW_male	0.25000
WSD	WW	WW	WSD_female	0.25000
WSD	WW	WW	WSD_male	0.25000
WW	SDSD	WW	WW_female	0.45000
WW	SDSD	WW	WW_male	0.45000
WW	SDSD	WW	WSD_female	0.05000
WW	SDSD	WW	WSD_male	0.05000
WW	WSD	WW	WW_female	0.47750
WW	WSD	WW	WW_male	0.47750
WW	WSD	WW	WSD_female	0.02250
WW	WSD	WW	WSD_male	0.02250
WW	WW	WW	WW_female	0.50000
WW	WW	WW	WW_male	0.50000

$~$

Function to determine the mating type frequencies of the population

$~$

This is a two-step process. First we determine the frequencies of each genotype in the mating population, then we calculate the frequencies of each potential mating combination between these genotypes.

$~$

Step 1 - finding the genotype frequencies in the mating population

While not measured in our study, we assume individuals with the genotype SDSD do not survive to adulthood (egg_adult_viability = 0), and that the viability of WSD individuals is unaffected.

# viability of each genotype

egg_adult_viability_WSD_female <- 1

egg_adult_viability_SDSD_female <- 0

egg_adult_viability_WSD_male <- 1

egg_adult_viability_SDSD_male <- 0

Next, we model the effect of the WSD genotype on a male’s mating success. We explore the parameter space where WSD males suffer costs when competing for copulations (S_precop$S_precop_WSD_male < 1), where the WSD genotype has the same mating success as the WW genotype (S_precop$S_precop_WSD_male = 1), and the parameter space where the WSD genotype is more successful than the WW genotype (S_precop$S_precop_WSD_male > 1). While it is possible that male choice for females may disadvantage SD carrying females, we have focused our efforts on measuring how precopulatory sexual selection affects the SD allele in males, as low quality males are more likely to go unmated than low quality females in the D. melanogaster polyandrous mating system.

Using these parameters, we implement selection, so that each genotype is represented in the mating combinations according to the product of its frequency, its viability and its mating success.

$~$

Step 2 - finding the frequency of all potential mating combinations in the mating population

We now calculate the expected frequencies of each possible ‘mating combination’ (i.e. the combination of the maternal and paternal genotypes) across the population. For singly-mated females, the frequency of each mating type is given by $F_iM_j(1 – p_j)$, where $F_i$ and $M_j$ are the female and male parental genotype frequencies, and $p_j$ is the probability of female remating following a first mating with a male of genotype $j$. Similarly, we found the expected frequencies of each possible mating type for females that mated with two males. We accomplished this via the formula $F_iM_jN_kp_j$, where $N_k$ represents the genotype frequency of the second male to mate.

find_mating_type_frequencies <- function(pop, prob_remating_SD, prob_remating_W){
  
  # Step 1
  
  pop$prop <- pop$prop * pop$egg_adult_viability * pop$S_precop # here's the selection step 
  pop$prop <- pop$prop / sum(pop$prop) # now restandardise so that the props sum to 1
  
  # Step 2
  
  # Calculate the frequency of each single-mating combination
  
  first_mating_types <- 
    expand_grid(mother = pop$type[str_detect(pop$type, "female")], # find all possible mother genotypes
                father1 = pop$type[!str_detect(pop$type, "female")]) %>% # find all possible father1 genotypes
    mutate(prop_matings = 0, remating_probability = 0) # make placeholder columns we will fill in below
  
  # find genotype frequencies calculated for each sex (mate choice means these will not necessarily be 50:50)
  
  female_frequencies <- pop %>% filter(str_detect(type, "female")) %>% mutate(prop = prop / sum(prop))
  male_frequencies <- pop %>% filter(!str_detect(type, "female")) %>% mutate(prop = prop / sum(prop))
  
  # This next step multiplies the proportion[i] of females with genotype[i] with the proportion[i] of males with genotype[i] 
  
  for(i in 1:nrow(first_mating_types)){
    first_mating_types$prop_matings[i] <- female_frequencies$prop[female_frequencies$type == first_mating_types$mother[i]] * 
      male_frequencies$prop[male_frequencies$type == first_mating_types$father1[i]]
  }
  
  # Now we incorporate female remating probabilities (SDSD males assumed to have same effect on females as WW males)
  
  first_mating_types$remating_probability <- ifelse(str_detect(first_mating_types$father1, "WSD"), prob_remating_SD, prob_remating_W)
  
  # Now lets calculate the frequency of each double-mating combination, with the end product being the mating_type_frequency of each mating possibility
  
  second_mating_types <- expand_grid(first_mating_types, father2 = c("none", unique(first_mating_types$father1))) %>%
    mutate(mating_type_freq = 0)
  
  # scale the proportion of matings calculated above by the likelihood that these females will mate once
  
  for(i in 1:nrow(second_mating_types)){
    if(second_mating_types$father2[i] == "none"){
      second_mating_types$mating_type_freq[i] <- 
        second_mating_types$prop_matings[i] * # Frequency of females with this value of father1
        (1 - second_mating_types$remating_probability[i])
    }
    
    # now calculate the frequency of mating tryads for the females that mate twice
    
    if(second_mating_types$father2[i] != "none"){
      second_mating_types$mating_type_freq[i] <- 
        second_mating_types$prop_matings[i] * # Frequency of females with this value of father1
        second_mating_types$remating_probability[i] *
        male_frequencies$prop[male_frequencies$type == second_mating_types$father2[i]] # pop. freq of the 2nd male type
    }
  }
  
  second_mating_types %>% # output
    select(mother, father1, father2, mating_type_freq) %>%
    # clean up the output by remving redundant info (makes it easier later)
    mutate(mother = str_remove_all(mother, "_female"),
           father1 = str_remove_all(father1, "_male"),
           father2 = str_remove_all(father2, "_male"))
}

$~$

Create a simple function to find the population-wide frequency of SD

$~$

calc_prop_SD <- function(pop){
  0.5 * sum(pop$prop[str_detect(pop$type, "WSD")]) + sum(pop$prop[str_detect(pop$type, "SDSD")])
}

$~$

Define the main simulation function

$~$

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

run_simulation <- function(row, parameters){
  
  print(paste("Doing row", row)) # this tells you which row in the parameter space is being modelled
  
  # Get the focal parameter space out of parameters[row], this makes it easier later
  
  generations <- parameters$generations[row]
  k <- parameters$k[row]
  P1cost <- parameters$P1cost[row]
  P2cost <- parameters$P2cost[row]
  normalP1 <- parameters$normalP1[row]
  prob_remating_SD <- parameters$prob_remating_SD[row]
  prob_remating_W <- parameters$prob_remating_W[row]
  egg_adult_viability_WSD_female <- parameters$egg_adult_viability_WSD_female[row]
  egg_adult_viability_SDSD_female <- parameters$egg_adult_viability_SDSD_female[row]
  egg_adult_viability_WSD_male <- parameters$egg_adult_viability_WSD_male[row]
  egg_adult_viability_SDSD_male <- parameters$egg_adult_viability_SDSD_male[row]
  S_precop_WSD_male <- parameters$S_precop_WSD_male[row]
  initial_freq_SD <- parameters$initial_freq_SD[row]
  
  
  # Make the initial population
  pop <- 
    tibble(
      type = c("WW_female",
               "WSD_female",
               "SDSD_female",
               "WW_male",
               "WSD_male",
               "SDSD_male"),
      prop = c(0.5 * (1 - initial_freq_SD) ^ 2,
               0.5 * (1 - initial_freq_SD) * initial_freq_SD * 2,
               0.5 * (initial_freq_SD^2),
               0.5 * (1 - initial_freq_SD) ^ 2,
               0.5 * (1 - initial_freq_SD) * initial_freq_SD * 2,
               0.5 * (initial_freq_SD^2)),
      egg_adult_viability = c(1, egg_adult_viability_WSD_female, egg_adult_viability_SDSD_female, 1,
                              egg_adult_viability_WSD_male, egg_adult_viability_SDSD_male),
      S_precop = c(1, 1, 1, 1, S_precop_WSD_male, 1)) %>% 
    arrange(type)
  
  # Make the mating table for this specific parameter space
  
  mating_table <- make_mating_table(k, normalP1, P1cost, P2cost)
  
  # Save the fitnesses of each genotype
  
  fitnesses <- pop %>% 
    select(type, egg_adult_viability, S_precop) 
  
  current_generation <- 1
  keep_going <- TRUE
  
  # Iterate over generations
  
  while(current_generation <= generations & keep_going){
    
    # Find the mating type frequencies, find the offspring frequencies, and renormalise the frequencies to sum to one every generation
    
    mating_type_frequencies <- find_mating_type_frequencies(
      pop = pop, 
      prob_remating_SD = prob_remating_SD, 
      prob_remating_W = prob_remating_W)
    
    pop <- left_join(mating_table, mating_type_frequencies,
                     by = c("mother", "father1", "father2")) %>%
      mutate(prop = mating_type_freq * zygote_freq) %>%
      group_by(type) %>%
      summarise(prop = sum(prop)) %>%
      left_join(fitnesses, by = "type")
    
    # Calculate the frequency of SD, quit early if SD fixes or goes extinct-ish
    prop_SD <- calc_prop_SD(pop)
    
    if(prop_SD > 0.99 | prop_SD < 0.0001) keep_going <- FALSE
    
    # Otherwise, increment generation counter by 1 and return to top of the 'while' loop
    
    current_generation <- current_generation + 1
  }
  
  pop # returns the final pop
  
}

$~$

Define the parameter space to investigate

$~$

resolution <- 25

gen <- 1000

initial_freq_SD <- 0.01

# updated with latest parameters

parameters <- rbind(
  expand_grid(
    generations = gen,
    k = c(estimated_k_SD5, estimated_k_SD72, estimated_k_SDMad),
    normalP1 = c(0.1, 0.5),
    P1cost = seq(0, 1, length = resolution),
    P2cost = seq(0, 1, length = resolution),
    prob_remating_W = estimated_prob_remating_W,
    prob_remating_SD = c(estimated_prob_remating_W, estimated_prob_remating_SD72, estimated_prob_remating_SD5),
    egg_adult_viability_WSD_female = egg_adult_viability_WSD_female,
    egg_adult_viability_SDSD_female = egg_adult_viability_SDSD_female,
    egg_adult_viability_WSD_male = egg_adult_viability_WSD_male,
    egg_adult_viability_SDSD_male = egg_adult_viability_SDSD_male,
    S_precop_WSD_male = c(0.8, 1, 1.2),
    initial_freq_SD = initial_freq_SD)) %>% 
  mutate(parameter_space_ID = row_number())

$~$

Checking that our simulation codes are consistent with previous analytical models

$~$

We can find SD allele frequencies for varying levels of segregation distortion if the only fitness cost associated with SD is homozygote lethality. This represents an upper bound for the frequency of SD alleles in our fully parameterised simulation.

Simplifying the simulation is also a useful exercise for comparing our model with previous models exploring the effect of homozygote lethality on segregation distorter frequencies. We find that our model reproduces the frequencies found using these previous models.

Bruck (1957) showed that homozygous lethal segregation distorting alleles (assuming no other fitness costs) reach an equilibrium frequency in adults at $q = \frac{1}{2} - \frac{\sqrt{k(1 - k)}}{2k}$, where $k$ is the strength of segregation distortion. We plug the three k values we observed in our pilot experiment into this equation and contrast the results with our simplified simulation results in Table S2 below.

$~$

Table S2: SD equilibrium frequencies calculated assuming varying levels of segregation distortion ($k$) and no fitness cost other than homozygote lethality associated with SD alleles. The table shows the equivalence between using our simulation code and the difference equation presented in Bruck (1957).

parameters_default <- rbind(
  expand_grid(
    generations = gen,
    k = c(0.5, estimated_k_SD5, estimated_k_SD72, estimated_k_SDMad),
    normalP1 = 0.5,
    P1cost = 0,
    P2cost = 0,
    prob_remating_W = 0,
    prob_remating_SD = 0,
    egg_adult_viability_WSD_female = egg_adult_viability_WSD_female,
    egg_adult_viability_SDSD_female = egg_adult_viability_SDSD_female,
    egg_adult_viability_WSD_male = egg_adult_viability_WSD_male,
    egg_adult_viability_SDSD_male = egg_adult_viability_SDSD_male,
    S_precop_WSD_male = 1,
    initial_freq_SD = initial_freq_SD)) %>% 
  mutate(parameter_space_ID = row_number())


if(!file.exists("simulation_results/output_defaults.rds")){
  output_defaults <- lapply(1:nrow(parameters_default), run_simulation, parameters = parameters_default)
  saveRDS(output_defaults, "simulation_results/output_defaults.rds")
} else{
  output_defaults <- read_rds("simulation_results/output_defaults.rds")
}

default_final_SD_props_among_preselection_zygotes <- sapply(output_defaults, calc_prop_SD)

default_final_SD_props_among_adults <- sapply(output_defaults, function(pop){
  
  pop$prop <- pop$prop * pop$egg_adult_viability # apply selection, then calculate allele freq of SD
  pop$prop <- pop$prop / sum(pop$prop) 
  calc_prop_SD(pop)
  
})


parameters_default %>%
  mutate(`Simulation SD equilibrium freq` = round(default_final_SD_props_among_adults, 3),
         `Bruck SD equilibrium freq` = round(1/2 - sqrt(k * (1- k)) / (2 * k), 3)) %>% 
  select(k, `Simulation SD equilibrium freq`, `Bruck SD equilibrium freq`) %>% 
  pander(split.cell = 40, split.table = Inf)

k	Simulation SD equilibrium freq	Bruck SD equilibrium freq
0.5	0.001	0
0.944	0.378	0.378
0.909	0.342	0.342
0.868	0.305	0.305

$~$

Run the simulation

$~$

# We run the simulation if the results don't already exist. The output is very large, which causes my mac to run out of memory before the simulation can finish. Therefore we break the simulation into 33 chunks to accommodate my 16gB of ram. After each chunk finishes, its output is saved to the hard disk and is then deleted from local memory, allowing the simulation to finish. We then combine the 33 chunks into a single object and save this as output.rds

if(!file.exists("simulation_results/output1.rds")){
  output1 <- lapply(1:1000, run_simulation, parameters = parameters)
  saveRDS(output1, "simulation_results/output1.rds")
}

rm(output1)
#gc()

if(!file.exists("simulation_results/output2.rds")){
  output2 <- lapply(1001:2000, run_simulation, parameters = parameters)
  saveRDS(output2, "simulation_results/output2.rds")
}

rm(output2)
#gc()

if(!file.exists("simulation_results/output3.rds")){
  output3 <- lapply(2001:3000, run_simulation, parameters = parameters)
  saveRDS(output3, "simulation_results/output3.rds")
}

rm(output3)
#gc()

if(!file.exists("simulation_results/output4.rds")){
  output4 <- lapply(3001:4000, run_simulation, parameters = parameters)
  saveRDS(output4, "simulation_results/output4.rds")
}

rm(output4)
#gc()
  
if(!file.exists("simulation_results/output5.rds")){
  output5 <- lapply(4001:5000, run_simulation, parameters = parameters)
  saveRDS(output5, "simulation_results/output5.rds")
}

rm(output5)
#gc()

if(!file.exists("simulation_results/output6.rds")){
  output6 <- lapply(5001:6000, run_simulation, parameters = parameters)
  saveRDS(output6, "simulation_results/output6.rds")
}

rm(output6)
#gc()

if(!file.exists("simulation_results/output7.rds")){
  output7 <- lapply(6001:7000, run_simulation, parameters = parameters)
  saveRDS(output7, "simulation_results/output7.rds")
}

rm(output7)
#gc()

if(!file.exists("simulation_results/output8.rds")){
  output8 <- lapply(7001:8000, run_simulation, parameters = parameters)
  saveRDS(output8, "simulation_results/output8.rds")
}

rm(output8)
#gc()

if(!file.exists("simulation_results/output9.rds")){
  output9 <- lapply(8001:9000, run_simulation, parameters = parameters)
  saveRDS(output9, "simulation_results/output9.rds")
}

rm(output9)
#gc()
  
if(!file.exists("simulation_results/output10.rds")){
  output10 <- lapply(9001:10000, run_simulation, parameters = parameters)
  saveRDS(output10, "simulation_results/output10.rds")
}

rm(output10)
#gc()

if(!file.exists("simulation_results/output11.rds")){
  output11 <- lapply(10001:11000, run_simulation, parameters = parameters)
  saveRDS(output11, "simulation_results/output11.rds")
}

rm(output11)
#gc()

if(!file.exists("simulation_results/output12.rds")){
  output12 <- lapply(11001:12000, run_simulation, parameters = parameters)
  saveRDS(output12, "simulation_results/output12.rds")
}

rm(output12)
#gc()

if(!file.exists("simulation_results/output13.rds")){
  output13 <- lapply(12001:13000, run_simulation, parameters = parameters)
  saveRDS(output13, "simulation_results/output13.rds")
}

rm(output13)
#gc()

if(!file.exists("simulation_results/output14.rds")){
  output14 <- lapply(13001:14000, run_simulation, parameters = parameters)
  saveRDS(output14, "simulation_results/output14.rds")
}

rm(output14)
#gc()
  
if(!file.exists("simulation_results/output15.rds")){
  output15 <- lapply(14001:15000, run_simulation, parameters = parameters)
  saveRDS(output15, "simulation_results/output15.rds")
}

rm(output15)
#gc()

if(!file.exists("simulation_results/output16.rds")){
  output16 <- lapply(15001:16000, run_simulation, parameters = parameters)
  saveRDS(output16, "simulation_results/output16.rds")
}

rm(output16)
#gc()

if(!file.exists("simulation_results/output17.rds")){
  output17 <- lapply(16001:17000, run_simulation, parameters = parameters)
  saveRDS(output17, "simulation_results/output17.rds")
}

rm(output17)
#gc()

if(!file.exists("simulation_results/output18.rds")){
  output18 <- lapply(17001:18000, run_simulation, parameters = parameters)
  saveRDS(output18, "simulation_results/output18.rds")
}

rm(output18)
#gc()

if(!file.exists("simulation_results/output19.rds")){
  output19 <- lapply(18001:19000, run_simulation, parameters = parameters)
  saveRDS(output19, "simulation_results/output19.rds")
}

rm(output19)
#gc()
  
if(!file.exists("simulation_results/output20.rds")){
  output20 <- lapply(19001:20000, run_simulation, parameters = parameters)
  saveRDS(output20, "simulation_results/output20.rds")
}

rm(output20)
#gc()


if(!file.exists("simulation_results/output21.rds")){
  output21 <- lapply(20001:21000, run_simulation, parameters = parameters)
  saveRDS(output21, "simulation_results/output21.rds")
}

rm(output21)
#gc()

if(!file.exists("simulation_results/output22.rds")){
  output22 <- lapply(21001:22000, run_simulation, parameters = parameters)
  saveRDS(output22, "simulation_results/output22.rds")
}

rm(output22)
#gc()

if(!file.exists("simulation_results/output23.rds")){
  output23 <- lapply(22001:23000, run_simulation, parameters = parameters)
  saveRDS(output23, "simulation_results/output23.rds")
}

rm(output23)
#gc()

if(!file.exists("simulation_results/output24.rds")){
  output24 <- lapply(23001:24000, run_simulation, parameters = parameters)
  saveRDS(output24, "simulation_results/output24.rds")
}

rm(output24)
#gc()
  
if(!file.exists("simulation_results/output25.rds")){
  output25 <- lapply(24001:25000, run_simulation, parameters = parameters)
  saveRDS(output25, "simulation_results/output25.rds")
}

rm(output25)
#gc()


if(!file.exists("simulation_results/output26.rds")){
  output26 <- lapply(25001:26000, run_simulation, parameters = parameters)
  saveRDS(output26, "simulation_results/output26.rds")
}

rm(output26)
#gc()

if(!file.exists("simulation_results/output27.rds")){
  output27 <- lapply(26001:27000, run_simulation, parameters = parameters)
  saveRDS(output27, "simulation_results/output27.rds")
}

rm(output27)
#gc()

if(!file.exists("simulation_results/output28.rds")){
  output28 <- lapply(27001:28000, run_simulation, parameters = parameters)
  saveRDS(output28, "simulation_results/output28.rds")
}

rm(output28)
#gc()

if(!file.exists("simulation_results/output29.rds")){
  output29 <- lapply(28001:29000, run_simulation, parameters = parameters)
  saveRDS(output29, "simulation_results/output29.rds")
}

rm(output29)
#gc()
  
if(!file.exists("simulation_results/output30.rds")){
  output30 <- lapply(29001:30000, run_simulation, parameters = parameters)
  saveRDS(output30, "simulation_results/output30.rds")
}

rm(output30)
#gc()

if(!file.exists("simulation_results/output31.rds")){
  output31 <- lapply(30001:31000, run_simulation, parameters = parameters)
  saveRDS(output31, "simulation_results/output31.rds")
}

rm(output31)
#gc()

if(!file.exists("simulation_results/output32.rds")){
  output32 <- lapply(31001:32000, run_simulation, parameters = parameters)
  saveRDS(output32, "simulation_results/output32.rds")
}

rm(output32)
#gc()

if(!file.exists("simulation_results/output33.rds")){
  output33 <- lapply(32001:33750, run_simulation, parameters = parameters)
  saveRDS(output33, "simulation_results/output33.rds")
}

# reload chunks from disk (output 33 was never deleted)
output1 <- readRDS("simulation_results/output1.rds")
output2 <- readRDS("simulation_results/output2.rds")
output3 <- readRDS("simulation_results/output3.rds")
output4 <- readRDS("simulation_results/output4.rds")
output5 <- readRDS("simulation_results/output5.rds")
output6 <- readRDS("simulation_results/output6.rds")
output7 <- readRDS("simulation_results/output7.rds")
output8 <- readRDS("simulation_results/output8.rds")
output9 <- readRDS("simulation_results/output9.rds")
output10 <- readRDS("simulation_results/output10.rds")
output11 <- readRDS("simulation_results/output11.rds")
output12 <- readRDS("simulation_results/output12.rds")
output13 <- readRDS("simulation_results/output13.rds")
output14 <- readRDS("simulation_results/output14.rds")
output15 <- readRDS("simulation_results/output15.rds")
output16 <- readRDS("simulation_results/output16.rds")
output17 <- readRDS("simulation_results/output17.rds")
output18 <- readRDS("simulation_results/output18.rds")
output19 <- readRDS("simulation_results/output19.rds")
output20 <- readRDS("simulation_results/output20.rds")
output21 <- readRDS("simulation_results/output21.rds")
output22 <- readRDS("simulation_results/output22.rds")
output23 <- readRDS("simulation_results/output23.rds")
output24 <- readRDS("simulation_results/output24.rds")
output25 <- readRDS("simulation_results/output25.rds")
output26 <- readRDS("simulation_results/output26.rds")
output27 <- readRDS("simulation_results/output27.rds")
output28 <- readRDS("simulation_results/output28.rds")
output29 <- readRDS("simulation_results/output29.rds")
output30 <- readRDS("simulation_results/output30.rds")
output31 <- readRDS("simulation_results/output31.rds")
output32 <- readRDS("simulation_results/output32.rds")
output33 <- readRDS("simulation_results/output33.rds")



if(!file.exists("simulation_results/output.rds")){
  output_combined <- c(output1, output2, output3, output4, output5, output6,
                       output7, output8, output9, output10, output11, output12,
                       output13, output14, output15, output16, output17, output18,
                       output19, output20, output21, output22, output23, output24,
                       output25, output26, output27, output28, output29, output30,
                       output31, output32, output33)
  saveRDS(output_combined, "simulation_results/output.rds")
} else{
  output <- readRDS("simulation_results/output.rds")
}

final_SD_props_among_preselection_zygotes <- sapply(output, calc_prop_SD)

final_SD_props_among_adults <- sapply(output, function(pop){
  
  pop$prop <- pop$prop * pop$egg_adult_viability # apply selection, then calculate allele freq of SD
  pop$prop <- pop$prop / sum(pop$prop) 
  calc_prop_SD(pop)
  
})

$~$

Create Figures 2 and S8

$~$

# Combine the SD frequencies with the parameter data and transform data to improve clarity

plotting_data <- parameters %>%
  mutate(prop_SD = final_SD_props_among_adults,
         P1_actual = (normalP1 -  P1cost * normalP1),
         P2_actual = ((1 - normalP1) - P2cost * (1 - normalP1)),
         remating_ratio = prob_remating_SD / prob_remating_W) %>% 
  mutate(remating_ratio = round(remating_ratio, digits = 1))

# Transform our empirical estimates to match the plotting data  

empirical_estimates_0.1 <-
  empirical_estimates %>% 
  mutate(W_remating_prob = 0.304,
         normalP1 = 0.1,
         P1_actual = (normalP1 -  P1cost * normalP1),
         lower_P1_actual = (normalP1 -  lower_P1cost * normalP1),
         upper_P1_actual = (normalP1 -  upper_P1cost * normalP1),
         P2_actual = ((1 - normalP1) - P2cost * (1 - normalP1)),
         lower_P2_actual = ((1 - normalP1) - lower_P2cost * (1 - normalP1)),
         upper_P2_actual = ((1 - normalP1) - upper_P2cost * (1 - normalP1)),
         remating_ratio = remating_prob / estimated_prob_remating_W) %>% 
  mutate(remating_ratio = round(remating_ratio, digits = 1))

# Now make Figure 2

(p1 <- 
    plotting_data %>% 
    rename(`Wildtype P1` = normalP1,
           `Relative risk of sperm competition` = remating_ratio,
           `Relative mating success` = S_precop_WSD_male) %>% 
    filter(k == 0.944, `Wildtype P1` == 0.1) %>% 
    ggplot(aes(x = P1_actual, y = P2_actual)) +
    geom_blank() +
    geom_raster(aes(fill = 100 * prop_SD)) + 
    stat_contour(aes(z = 100 * prop_SD), colour = "black", binwidth = 10) +
    stat_contour(aes(z = 100 * prop_SD), colour = "black", breaks = 8, linetype = 2) +
    scale_fill_carto_c(type = "sequential", palette = "BluYl", direction = 1, name = "% SD at equilibrium") +
    geom_errorbar(data = empirical_estimates_0.1,
                  aes(x = P1_actual, ymin = lower_P2_actual, ymax = upper_P2_actual),
                  width = 0.003, colour = "grey11", alpha = 0.8) +
    geom_errorbarh(data = empirical_estimates_0.1,
                   aes(y = P2_actual, xmin = lower_P1_actual, xmax = upper_P1_actual),
                   colour = "grey11", alpha = 0.8, height = 0.03) +
    new_scale_fill() + # geoms below use a new scale
    geom_point(data = empirical_estimates_0.1 %>% 
                 rename(`SD-variant` = SD),
               aes(x = P1_actual, y = P2_actual, fill = `SD-variant`), 
               pch = 21, colour = "grey11", size = 3) +
    scale_fill_manual(values = c("SD-Mad" = "#EDAD08", "SD-72" = "#CC503E", "SD-5" = "#994E95")) +
    facet_wrap(`Relative risk of sperm competition` ~ `Relative mating success`, 
               scales = "free", nrow = 3, strip.position = c("top"),
               labeller = label_glue('({.l}) Relative risk of sperm competition: {`Relative risk of sperm competition`}\nRelative mating success: {`Relative mating success`}')) +
    labs(x = "Prop. offspring sired by SD/+ male in the first mating role (P1)",
         y = "Prop. offspring sired by SD/+ male in the second mating role (P2)") +
    scale_x_continuous(expand = c(0, 0)) + 
    scale_y_continuous(expand = c(0, 0)) + # labels = c(0, 25, 50, 75, 90)) +
    theme(panel.border = element_rect(fill = NA, colour = "black", size = .8),
          strip.background = element_rect(colour = "black", fill = "Aliceblue", size = .8))
)

Version	Author	Date
2856826	tkeaney	2021-09-01
7a63896	tkeaney	2021-08-31
e345048	tkeaney	2021-08-30
35cfc61	tkeaney	2021-08-07
f30258a	tkeaney	2021-05-25
ddaa2d1	tkeaney	2021-05-24

Figure 2: Predicted equilibrium frequency of the SD allele, calculated from the population genetic model. The plot depicts the interaction between the P1 (x-axis) and P2 (y-axis) costs suffered by SD/+ males in their effects on the equilibrium frequency of SD (shown by the colour scale and 10% contour lines). The dashed line shows an equilibrium frequency of 8%, the upper estimate for SD alleles in natural populations. The plot is split into nine panels, with varying levels of SD/+ male mating success ($S_{precop}$) across the columns and an increasing risk of sperm competition caused by a female remating after first mating to an SD/+ male, $p_{SD/+}$, down the rows (values correspond to the risk of sperm competition we estimated in Experiment 2). The three points (with associated 95% credible intervals) in each panel show where males carrying each SD variant fall in the figure’s parameter space. In the parameter space presented, $k$ = 0.944, $P1_{normal}$ = 0.1, and SD homozygotes are non-viable.

$~$

We measured P1 and P2 with a set interval of four days between matings. While we found that many females did not remate within the three-hour allotted mating period, many studies show that a low % of females will remate much faster than this, and that the interval between matings affects P1 and P2. Matings that occur closer temporally result in higher P1 values (e.g. Laturney, Eijk and Billeter, 2018, Evolution Letters. In addition, many populations/species do not exhibit the extreme second male precedence we observed in our study. Therefore, we also modelled the scenario where there was no effect of mating order on fertilisation success, that is $P1_{normal}$ = 0.5. The results are shown in Figure S8 below.

# Transform our empirical estimates to match the plotting data  

empirical_estimates_0.5 <-
  empirical_estimates %>% 
  mutate(W_remating_prob = 0.30,
         normalP1 = 0.5,
         P1_actual = (normalP1 -  P1cost * normalP1),
         lower_P1_actual = (normalP1 -  lower_P1cost * normalP1),
         upper_P1_actual = (normalP1 -  upper_P1cost * normalP1),
         P2_actual = ((1 - normalP1) - P2cost * (1 - normalP1)),
         lower_P2_actual = ((1 - normalP1) - lower_P2cost * (1 - normalP1)),
         upper_P2_actual = ((1 - normalP1) - upper_P2cost * (1 - normalP1)),
         remating_ratio = remating_prob / estimated_prob_remating_W) %>% 
  mutate(remating_ratio = round(remating_ratio, digits = 1))

# Make Figure S8

(p2 <-
    plotting_data %>% 
    rename(`Wildtype P1` = normalP1,
           `Relative risk of sperm competition` = remating_ratio,
           `Relative mating success` = S_precop_WSD_male) %>% 
    filter(k == 0.944, `Wildtype P1` == 0.5) %>% 
    ggplot(aes(x = P1_actual, y = P2_actual)) +
    geom_blank() +
    geom_raster(aes(fill = 100 * prop_SD)) + 
    stat_contour(aes(z = 100 * prop_SD), colour = "black", binwidth = 10) +
    stat_contour(aes(z = 100 * prop_SD), colour = "black", breaks = 8, linetype = 2) +
    scale_fill_carto_c(type = "sequential", palette = "BluYl", direction = 1, name = "% SD at equilibrium") +
    geom_errorbar(data = empirical_estimates_0.5,
                  aes(x = P1_actual, ymin = lower_P2_actual, ymax = upper_P2_actual),
                  width = 0.02, colour = "grey11", alpha = 0.8) +
    geom_errorbarh(data = empirical_estimates_0.5,
                   aes(y = P2_actual, xmin = lower_P1_actual, xmax = upper_P1_actual),
                   colour = "grey11", alpha = 0.8, height = 0.02) +
    new_scale_fill() + # geoms below use a new scale
    geom_point(data = empirical_estimates_0.5 %>% 
                 rename(`SD-variant` = SD),
               aes(x = P1_actual, y = P2_actual, fill = `SD-variant`), 
               pch = 21, colour = "grey11", size = 3) +
    scale_fill_manual(values = c("SD-Mad" = "#EDAD08", "SD-72" = "#CC503E", "SD-5" = "#994E95")) +
    facet_wrap(`Relative risk of sperm competition` ~ `Relative mating success`, 
               scales = "free", nrow = 3, strip.position = c("top"),
               labeller = label_glue('({.l}) Relative risk of sperm competition: {`Relative risk of sperm competition`}\nRelative mating success: {`Relative mating success`}')) +
    labs(x = "Prop. offspring sired by SD/+ male in the first mating role (P1)",
         y = "Prop. offspring sired by SD/+ male in the second mating role (P2)") +
    scale_x_continuous(expand = c(0, 0)) + 
    scale_y_continuous(expand = c(0, 0)) + # labels = c(0, 25, 50, 75, 90)) +
    theme(panel.border = element_rect(fill = NA, colour = "black", size = .8),
          strip.background = element_rect(colour = "black", fill = "Aliceblue", size = .8))
)

Version	Author	Date
2856826	tkeaney	2021-09-01
7a63896	tkeaney	2021-08-31

Figure S8: Predicted equilibrium frequency of the SD allele, calculated from the population genetic model. The plot depicts the interaction between the P1 (x-axis) and P2 (y-axis) costs suffered by SD/+ males in their effects on the equilibrium frequency of SD (shown by the colour scale and 10% contour lines). The dashed line shows an equilibrium frequency of 8%, the upper estimate for SD alleles in natural populations. The plot is split into nine panels, with varying levels of SD/+ male mating success across the rows and increasing likelihoods of a female remating after first mating to an SD/+ male, $p_{SD/+}$, down the columns. The three points (with associated 95% credible intervals) in each panel show where males carrying each SD variant fall in the figure’s parameter space. In the parameter space presented here, $k$ = 0.944 (our highest estimated value), $P1_{normal}$ = 0.5, SD homozygotes are non-viable and SD heterozygotes suffer no fitness costs outside of mating success and sperm competition.

sessionInfo()

R version 4.1.0 (2021-05-18)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS Big Sur 10.16

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.1/Resources/lib/libRblas.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.1/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] ggnewscale_0.4.5          stickylabeller_0.0.0.9100
 [3] pander_0.6.4              kableExtra_1.3.4         
 [5] ggpubr_0.4.0              rcartocolor_2.0.0        
 [7] patchwork_1.1.1           forcats_0.5.1            
 [9] stringr_1.4.0             dplyr_1.0.7              
[11] purrr_0.3.4               readr_2.0.0              
[13] tidyr_1.1.3               tibble_3.1.3             
[15] ggplot2_3.3.5             tidyverse_1.3.1          
[17] workflowr_1.6.2          

loaded via a namespace (and not attached):
 [1] fs_1.5.0          lubridate_1.7.10  webshot_0.5.2     httr_1.4.2       
 [5] rprojroot_2.0.2   tools_4.1.0       backports_1.2.1   bslib_0.2.5.1    
 [9] utf8_1.2.2        R6_2.5.0          DBI_1.1.1         colorspace_2.0-2 
[13] withr_2.4.2       tidyselect_1.1.1  curl_4.3.2        compiler_4.1.0   
[17] git2r_0.28.0      cli_3.0.1         rvest_1.0.1       xml2_1.3.2       
[21] isoband_0.2.5     labeling_0.4.2    sass_0.4.0        scales_1.1.1     
[25] systemfonts_1.0.2 digest_0.6.27     foreign_0.8-81    rmarkdown_2.10   
[29] svglite_2.0.0     rio_0.5.27        pkgconfig_2.0.3   htmltools_0.5.1.1
[33] highr_0.9         dbplyr_2.1.1      rlang_0.4.11      readxl_1.3.1     
[37] rstudioapi_0.13   farver_2.1.0      jquerylib_0.1.4   generics_0.1.0   
[41] jsonlite_1.7.2    zip_2.2.0         car_3.0-11        magrittr_2.0.1   
[45] Rcpp_1.0.7        munsell_0.5.0     fansi_0.5.0       abind_1.4-5      
[49] lifecycle_1.0.0   stringi_1.7.3     whisker_0.4       yaml_2.2.1       
[53] carData_3.0-4     grid_4.1.0        promises_1.2.0.1  crayon_1.4.1     
[57] haven_2.4.3       hms_1.1.0         knitr_1.33        pillar_1.6.2     
[61] ggsignif_0.6.2    reprex_2.0.1      glue_1.4.2        evaluate_0.14    
[65] data.table_1.14.0 modelr_0.1.8      vctrs_0.3.8       tzdb_0.1.2       
[69] httpuv_1.6.1      cellranger_1.1.0  gtable_0.3.0      assertthat_0.2.1 
[73] xfun_0.25         openxlsx_4.2.4    broom_0.7.9       rstatix_0.7.0    
[77] later_1.2.0       viridisLite_0.4.0 ellipsis_0.3.2

Sexual selection can partly explain low frequencies of Segregation Distorter alleles

Population genetic model

Thomas Keaney, Theresa Jones and Luke Holman