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Rmd fd05fc4 Dave Tang 2026-02-16 Simulating evolution

Introduction

Population genetics studies how allele frequencies change in populations over time. The key forces driving these changes are genetic drift (random sampling), natural selection (differential fitness), and mutation (introduction of new variants). Analytical solutions exist for simple models, but simulation is a powerful way to build intuition about how these forces interact, especially in finite populations where randomness plays a large role.

Here, we will simulate evolution using only base R. We will start with drift alone, then layer on selection and mutation to see how each force shapes allele frequency trajectories.

Genetic drift

Genetic drift is the random change in allele frequencies that occurs because populations are finite. Imagine a population of N diploid individuals (so 2N gene copies). Each generation, the next generation is formed by randomly sampling 2N alleles from the current pool; like drawing marbles from a bag with replacement.

The function sim_drift() below simulates this process. It takes three parameters:

  • p0 - the initial frequency of allele A (between 0 and 1)
  • N - the number of diploid individuals in the population (so there are 2N gene copies)
  • generations - the number of generations to simulate

Each generation, the function draws the number of A alleles in the next generation from a binomial distribution: rbinom(1, size = 2*N, prob = p), where p is the current frequency of A. This is equivalent to each of the 2N gene copies in the new generation independently choosing to be A with probability p - the same as sampling with replacement from the current allele pool. The new frequency is then the count divided by 2N. This random binomial sampling is what produces genetic drift: even if the “true” probability of A is p, the realised frequency will fluctuate due to finite sampling.

Note that this model (the Wright-Fisher model) operates on the allele pool directly and does not track individual genotypes (AA, Aa, aa). It only tracks the frequency of allele A; the frequency of the other allele (a) is implicitly 1 - p. This means a starting frequency of p0 = 0.5 does not imply all individuals are heterozygous; it simply means half the allele copies in the population are A, which could arise from any combination of genotypes that produces that overall frequency. The binomial sampling each generation implicitly assumes random mating (Hardy-Weinberg).

Track the frequency of allele “A” (as opposed to “a”) over 100 generations.

set.seed(1984)

sim_drift <- function(p0, N, generations) {
  freq <- numeric(generations + 1)
  freq[1] <- p0
  for (g in seq_len(generations)) {
    n_A <- rbinom(1, size = 2 * N, prob = freq[g])
    freq[g + 1] <- n_A / (2 * N)
  }
  freq
}

generations <- 100
p0 <- 0.5
N <- 50

# Run 10 replicate populations
n_reps <- 10
cols <- rainbow(n_reps)

plot(0:generations, sim_drift(p0, N, generations),
     type = "n", ylim = c(0, 1),
     xlab = "Generation", ylab = "Frequency of A",
     main = paste("Genetic drift: N =", N))
abline(h = p0, lty = 2, col = "grey50")

for (i in seq_len(n_reps)) {
  lines(0:generations, sim_drift(p0, N, generations), col = cols[i])
}

Each coloured line is an independent replicate population. They all start at the same frequency (0.5) but wander apart due to random sampling. Some may reach 0 (allele A lost) or 1 (allele A fixed).

Population size matters

Smaller populations experience stronger drift. Let’s compare N = 20 versus N = 500.

par(mfrow = c(1, 2))

for (N_val in c(20, 500)) {
  plot(0:generations, rep(NA, generations + 1),
       type = "n", ylim = c(0, 1),
       xlab = "Generation", ylab = "Frequency of A",
       main = paste("N =", N_val))
  abline(h = p0, lty = 2, col = "grey50")
  for (i in seq_len(n_reps)) {
    lines(0:generations, sim_drift(p0, N_val, generations), col = cols[i])
  }
}

par(mfrow = c(1, 1))

With N = 20 the trajectories are wild and several reach fixation or loss within 100 generations. With N = 500 the trajectories stay much closer to the starting frequency.

Drift leads to fixation

A fundamental result of drift theory is that the probability of an allele eventually reaching fixation equals its starting frequency. If allele A starts at frequency p, then across many replicate populations, the fraction that fix for A should be approximately p.

This is because drift is an unbiased process; on average, the allele frequency does not change from one generation to the next. The binomial sampling is centred on the current frequency, so drift has no preferred direction. However, once an allele reaches a frequency of 0 or 1, it stays there permanently (there is no mutation to bring it back in this model). These are absorbing states. So while drift does not favour either allele, every population will eventually wander into one of these absorbing states. The expected time to fixation scales with population size; specifically, the average time for a neutral allele to fix (conditional on fixation occurring) is approximately \(4N\) generations.

The fact that fixation probability equals starting frequency also has a useful corollary: a new mutation present as a single copy in a diploid population of size N has a fixation probability of \(1/(2N)\). In a population of 1,000 individuals, any given neutral mutation has only a 0.05% chance of eventually reaching fixation - the vast majority are lost to drift.

Let’s test the fixation probability result by running 1,000 replicate populations and letting them run until fixation (or a maximum of 5N generations, which is usually enough).

set.seed(42)

p0 <- 0.3
N <- 50
max_gen <- 5 * 2 * N  # 5 * 2N is a typical timescale for fixation
n_reps <- 1000

final_freq <- replicate(n_reps, {
  traj <- sim_drift(p0, N, max_gen)
  traj[max_gen + 1]
})

fixed_A <- sum(final_freq == 1)
lost_A <- sum(final_freq == 0)
still_seg <- n_reps - fixed_A - lost_A

barplot(
  c("Fixed (A)" = fixed_A, "Lost (a)" = lost_A, "Segregating" = still_seg),
  col = c("steelblue", "tomato", "grey70"),
  main = paste0("Fixation outcomes (p0 = ", p0, ", N = ", N, ", ", n_reps, " reps)"),
  ylab = "Number of populations"
)
abline(h = n_reps * p0, lty = 2)
legend("topright", legend = paste("Expected fixations:", n_reps * p0), lty = 2, bty = "n")

The number of populations that fixed for A is close to 273, compared to the expected value of 300. The remaining 8 populations haven’t reached fixation yet but would eventually do so given more time.

Natural selection

Natural selection occurs when different alleles confer different fitness. We can model this by giving allele A a selective advantage s. In a simple haploid model, the probability that an A allele is sampled into the next generation is proportional to 1 + s relative to allele a (fitness 1).

The function sim_selection() below extends sim_drift() by adding a selection step before the binomial sampling. It takes an additional parameter s, the selection coefficient, which represents the fitness advantage of allele A. The key line is:

\[p_{\text{sel}} = \frac{p \cdot (1 + s)}{p \cdot (1 + s) + (1 - p)}\]

This is a fitness-weighted frequency. The numerator is the total fitness contribution of A alleles (frequency p times fitness 1 + s) and the denominator is the total fitness of the entire population (A’s contribution plus a’s contribution at fitness 1). The result p_sel is always slightly higher than p when s > 0, giving A a systematic push upward each generation. This adjusted frequency is then passed to rbinom() instead of the raw frequency, so selection biases the sampling while drift still adds randomness around that biased expectation.

For example, if p = 0.5 and s = 0.05, then p_sel = (0.5 * 1.05) / (0.5 * 1.05 + 0.5) = 0.525 / 1.025 ≈ 0.512. The shift is small in any single generation, but it accumulates over time.

set.seed(1984)

sim_selection <- function(p0, N, generations, s) {
  freq <- numeric(generations + 1)
  freq[1] <- p0
  for (g in seq_len(generations)) {
    p <- freq[g]
    # Fitness-weighted frequency
    p_sel <- (p * (1 + s)) / (p * (1 + s) + (1 - p))
    n_A <- rbinom(1, size = 2 * N, prob = p_sel)
    freq[g + 1] <- n_A / (2 * N)
  }
  freq
}

generations <- 100
p0 <- 0.1
N <- 100
s <- 0.05

plot(0:generations, rep(NA, generations + 1),
     type = "n", ylim = c(0, 1),
     xlab = "Generation", ylab = "Frequency of A",
     main = "Drift only (grey) vs. Selection s = 0.05 (blue)")

# Drift only (grey)
for (i in 1:10) {
  lines(0:generations, sim_drift(p0, N, generations), col = "grey70")
}

# Selection (blue)
for (i in 1:10) {
  lines(0:generations, sim_selection(p0, N, generations, s), col = "steelblue")
}

abline(h = p0, lty = 2)
legend("topleft", legend = c("Drift only", "Selection (s = 0.05)"),
       col = c("grey70", "steelblue"), lwd = 2, bty = "n")

With selection, allele A tends to increase in frequency over time. Drift still introduces randomness, but the selective advantage biases the trajectory upward. Some drift-only replicates may coincidentally rise, but on average only the selected allele shows a consistent upward trend.

Strength of selection versus drift

Selection is effective when s is much larger than 1/(2N). When s is on the order of 1/(2N) or smaller, drift dominates and selection is essentially neutral. The intuition is that drift causes random frequency changes of order \(1/\sqrt{2N}\) each generation. If the deterministic push from selection (s) is much smaller than this random noise, the allele behaves as though it were neutral - selection simply cannot be “heard” above the stochastic fluctuations.

The critical threshold is \(s \approx 1/(2N)\). This defines two regimes:

  • Effectively neutral (\(s \ll 1/(2N)\)): the allele’s fate is governed almost entirely by drift. Its fixation probability is close to the neutral expectation of \(p\).
  • Effectively selected (\(s \gg 1/(2N)\)): selection dominates and the allele is driven toward fixation (if beneficial) or loss (if deleterious) in a largely predictable way.

For example, with N = 50 the threshold is \(1/(2 \times 50) = 0.01\). A selection coefficient of s = 0.01 would sit right at the boundary, while s = 0.1 (ten times larger) would be firmly in the selection-dominated regime. The simulation below compares these two cases.

set.seed(42)

N <- 50
p0 <- 0.5
generations <- 200

par(mfrow = c(1, 2))

# Weak selection: s ~ 1/(2N)
s_weak <- 1 / (2 * N)
plot(0:generations, rep(NA, generations + 1), type = "n", ylim = c(0, 1),
     xlab = "Generation", ylab = "Frequency of A",
     main = paste0("Weak selection (s = ", round(s_weak, 4), ")"))
for (i in 1:10) {
  lines(0:generations, sim_selection(p0, N, generations, s_weak), col = "steelblue", lwd = 0.8)
}
abline(h = p0, lty = 2)

# Strong selection: s >> 1/(2N)
s_strong <- 0.1
plot(0:generations, rep(NA, generations + 1), type = "n", ylim = c(0, 1),
     xlab = "Generation", ylab = "Frequency of A",
     main = paste0("Strong selection (s = ", s_strong, ")"))
for (i in 1:10) {
  lines(0:generations, sim_selection(p0, N, generations, s_strong), col = "tomato", lwd = 0.8)
}
abline(h = p0, lty = 2)

par(mfrow = c(1, 1))

With weak selection (left panel), the trajectories look almost indistinguishable from pure drift. With strong selection (right panel), allele A rapidly fixes in most replicates.

Mutation

Mutation introduces new genetic variation. We model this with two rates: mu (A mutates to a) and nu (a mutates to A). After selection (or drift) determines the allele frequency, mutation adjusts it:

\[p' = p \cdot (1 - \mu) + (1 - p) \cdot \nu\]

This means some A alleles become a (at rate mu) and some a alleles become A (at rate nu). The first term, \(p \cdot (1 - \mu)\), is the fraction of A alleles that survive without mutating. The second term, \((1 - p) \cdot \nu\), is the fraction of a alleles that mutate into A. Together they give the post-mutation frequency of A.

The function sim_drift_mutation() below adds two parameters to the drift model:

  • mu - the per-generation probability that an A allele mutates to a (forward mutation rate)
  • nu - the per-generation probability that an a allele mutates to A (back mutation rate)

Each generation, the mutation formula is applied first to compute an adjusted frequency p_mut, which is then passed to rbinom() for the drift step. This ordering means mutation shifts the expected frequency slightly before random sampling introduces noise.

An important consequence of bidirectional mutation is that the system has a stable equilibrium. Setting \(p' = p\) in the mutation equation and solving gives \(\hat{p} = \nu / (\mu + \nu)\). When mu and nu are equal, the equilibrium is 0.5. Unlike pure drift, mutation prevents the allele from ever permanently fixing or being lost - if A drifts to 0, the back mutation rate nu reintroduces it; if A drifts to 1, the forward rate mu erodes it. Typical mutation rates in biology are very small (\(10^{-6}\) to \(10^{-9}\) per base per generation), but we use larger values here so the effect is visible over a tractable number of generations.

set.seed(1984)

sim_drift_mutation <- function(p0, N, generations, mu, nu) {
  freq <- numeric(generations + 1)
  freq[1] <- p0
  for (g in seq_len(generations)) {
    p <- freq[g]
    # Mutation
    p_mut <- p * (1 - mu) + (1 - p) * nu
    # Drift
    n_A <- rbinom(1, size = 2 * N, prob = p_mut)
    freq[g + 1] <- n_A / (2 * N)
  }
  freq
}

N <- 50
p0 <- 0.5
generations <- 500
mu <- 0.01
nu <- 0.01

# Expected equilibrium frequency: nu / (mu + nu)
p_eq <- nu / (mu + nu)

plot(0:generations, rep(NA, generations + 1), type = "n", ylim = c(0, 1),
     xlab = "Generation", ylab = "Frequency of A",
     main = "Drift + Mutation")

for (i in 1:10) {
  lines(0:generations, sim_drift_mutation(p0, N, generations, mu, nu),
        col = adjustcolor("steelblue", alpha.f = 0.6))
}

abline(h = p_eq, lty = 2, col = "red", lwd = 2)
legend("topright", legend = paste("Expected equilibrium:", p_eq),
       lty = 2, col = "red", lwd = 2, bty = "n")

Mutation prevents fixation! Even in a small population, the allele frequency fluctuates around the equilibrium value \(\hat{p} = \nu / (\mu + \nu)\). With equal forward and backward mutation rates, this equilibrium is 0.5.

Mutation maintains variation

Without mutation, drift will eventually fix or lose alleles. With mutation, polymorphism is maintained. This is a fundamental distinction: drift alone is a process that destroys genetic variation (by pushing alleles to fixation or loss), while mutation is the ultimate source of new variation.

In the drift-only model, 0 and 1 are absorbing states - once an allele is fixed or lost, the population stays there forever. With mutation, these states are no longer absorbing. If A is lost (frequency = 0), the back mutation term \((1 - p) \cdot \nu = \nu\) ensures that A is reintroduced at a low rate each generation. Similarly, if A reaches fixation (frequency = 1), the forward mutation term \(p \cdot (1 - \mu) = 1 - \mu\) pulls it back below 1. The allele frequency therefore bounces around the equilibrium indefinitely rather than settling at a boundary.

The balance between drift and mutation determines how much variation is maintained. The key parameter is \(\theta = 4N\mu\) (for a diploid population), known as the population-scaled mutation rate. When \(\theta\) is large (large population or high mutation rate), allele frequencies cluster tightly around the equilibrium. When \(\theta\) is small, drift dominates and frequencies swing widely, occasionally approaching 0 or 1 before mutation pulls them back. The simulation below uses N = 30 and mu = nu = 0.01, giving \(\theta = 4 \times 30 \times 0.01 = 1.2\), which produces noticeable fluctuations but prevents fixation.

set.seed(42)

N <- 30
p0 <- 0.5
generations <- 300

par(mfrow = c(1, 2))

# Drift only
plot(0:generations, rep(NA, generations + 1), type = "n", ylim = c(0, 1),
     xlab = "Generation", ylab = "Frequency of A",
     main = "Drift only (N = 30)")
for (i in 1:10) {
  lines(0:generations, sim_drift(p0, N, generations),
        col = adjustcolor("grey40", alpha.f = 0.6))
}

# Drift + mutation
plot(0:generations, rep(NA, generations + 1), type = "n", ylim = c(0, 1),
     xlab = "Generation", ylab = "Frequency of A",
     main = "Drift + Mutation (N = 30)")
for (i in 1:10) {
  lines(0:generations, sim_drift_mutation(p0, N, generations, 0.01, 0.01),
        col = adjustcolor("steelblue", alpha.f = 0.6))
}
abline(h = 0.5, lty = 2, col = "red")

par(mfrow = c(1, 1))

In the drift-only case (left), most populations hit 0 or 1 and stay there. With mutation (right), no population can stay fixed because mutation keeps reintroducing the lost allele.

Bringing it together

Now let’s combine all three forces - drift, selection, and mutation - in a single simulation. We give allele A a selective advantage and include bidirectional mutation.

sim_evolution <- function(p0, N, generations, s = 0, mu = 0, nu = 0) {
  freq <- numeric(generations + 1)
  freq[1] <- p0
  for (g in seq_len(generations)) {
    p <- freq[g]
    # Selection
    p_sel <- (p * (1 + s)) / (p * (1 + s) + (1 - p))
    # Mutation
    p_mut <- p_sel * (1 - mu) + (1 - p_sel) * nu
    # Drift
    n_A <- rbinom(1, size = 2 * N, prob = p_mut)
    freq[g + 1] <- n_A / (2 * N)
  }
  freq
}

Let’s compare four scenarios side by side.

set.seed(1984)

N <- 100
p0 <- 0.1
generations <- 300
n_reps <- 10

scenarios <- list(
  "Drift only" = list(s = 0, mu = 0, nu = 0),
  "Selection only (s = 0.03)" = list(s = 0.03, mu = 0, nu = 0),
  "Mutation only (mu = nu = 0.005)" = list(s = 0, mu = 0.005, nu = 0.005),
  "All forces combined" = list(s = 0.03, mu = 0.005, nu = 0.005)
)

par(mfrow = c(2, 2))

for (name in names(scenarios)) {
  params <- scenarios[[name]]
  plot(0:generations, rep(NA, generations + 1), type = "n", ylim = c(0, 1),
       xlab = "Generation", ylab = "Frequency of A", main = name)
  abline(h = p0, lty = 2, col = "grey50")
  for (i in seq_len(n_reps)) {
    traj <- sim_evolution(p0, N, generations,
                          s = params$s, mu = params$mu, nu = params$nu)
    lines(0:generations, traj, col = adjustcolor("steelblue", alpha.f = 0.5))
  }
}

par(mfrow = c(1, 1))

Some observations:

  • Drift only: allele frequencies wander randomly; some replicates drift toward fixation or loss.
  • Selection only: allele A rises toward fixation thanks to its fitness advantage, though drift adds variability in the path.
  • Mutation only: frequencies fluctuate around the equilibrium \(\nu / (\mu + \nu) = 0.5\) and fixation is prevented.
  • All forces combined: selection pushes A upward, but mutation prevents complete fixation - the frequency equilibrates at a value determined by the balance between selection and mutation.

These simulations illustrate a core insight of population genetics: evolution is not a single force but an interplay of deterministic forces (selection, mutation) and stochastic processes (drift). The relative strengths of these forces, governed by population size and the magnitudes of s, mu, and nu, determine the evolutionary outcome.


sessionInfo()
R version 4.5.2 (2025-10-31)
Platform: x86_64-pc-linux-gnu
Running under: Ubuntu 24.04.4 LTS

Matrix products: default
BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3 
LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so;  LAPACK version 3.12.0

locale:
 [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
 [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
 [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
 [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
 [9] LC_ADDRESS=C               LC_TELEPHONE=C            
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       

time zone: Etc/UTC
tzcode source: system (glibc)

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

other attached packages:
[1] workflowr_1.7.2

loaded via a namespace (and not attached):
 [1] vctrs_0.7.1       httr_1.4.8        cli_3.6.5         knitr_1.51       
 [5] rlang_1.1.7       xfun_0.56         stringi_1.8.7     otel_0.2.0       
 [9] processx_3.8.6    promises_1.5.0    jsonlite_2.0.0    glue_1.8.0       
[13] rprojroot_2.1.1   git2r_0.36.2      htmltools_0.5.9   httpuv_1.6.16    
[17] ps_1.9.1          sass_0.4.10       rmarkdown_2.30    jquerylib_0.1.4  
[21] tibble_3.3.1      evaluate_1.0.5    fastmap_1.2.0     yaml_2.3.12      
[25] lifecycle_1.0.5   whisker_0.4.1     stringr_1.6.0     compiler_4.5.2   
[29] fs_1.6.6          pkgconfig_2.0.3   Rcpp_1.1.1        rstudioapi_0.18.0
[33] later_1.4.6       digest_0.6.39     R6_2.6.1          pillar_1.11.1    
[37] callr_3.7.6       magrittr_2.0.4    bslib_0.10.0      tools_4.5.2      
[41] cachem_1.1.0      getPass_0.2-4