Experiment 03Genetic DriftDemographic Modeling
Genetic Drift, Founder Effects & Effective Population Size ($N_e$) in Finite Demes: Wright-Fisher Simulation
Duration: 4 Hours • Computational & Physical Deme Lab
🎯 Experimental Aim & Learning Objectives
To investigate non-directional stochastic changes in allele frequencies caused by random sampling error in finite populations (Wright-Fisher model); measure the time to allele fixation and allele loss as a function of population size ($N = 10$ vs $N = 500$); observe decay of expected heterozygosity ($H_t$); and calculate Wright’s effective population size ($N_e$) under unequal sex ratios and demographic bottlenecks.
🔬 Theoretical Foundation & Algorithmic Principles
In finite populations, chance sampling of gametes produces random genetic drift, leading to non-adaptive evolutionary change independent of natural selection:
- Probability of Fixation & Fixation Time: The probability that a neutral allele will eventually become fixed equals its current frequency ($P(text{fixation}) = p_0$). In a diploid population of size $N$, the mean time to fixation for a newly arisen neutral mutation is:
$$bar{t}_{text{fixation}} = 4N text{ generations}$$
whereas a lost allele takes on average $bar{t}_{text{loss}} = 2 left(frac{N}{p}right) ln(2N)$ generations. - Decay of Heterozygosity: Drift continuously erodes genetic variation at a rate inversely proportional to population size:
$$H_t = H_0 left(1 – frac{1}{2N_e}right)^t$$ - Wright’s Effective Population Size ($N_e$): The size of an idealized Fisher-Wright population that would experience the same rate of genetic drift or inbreeding as the actual population:
• Unequal Sex Ratio: $N_e = frac{4 N_m N_f}{N_m + N_f}$ (A population of 2 males and 98 females has an $N_e approx 7.84$, not 100).
• Demographic Bottlenecks: Over $t$ generations with fluctuating sizes $N_1, N_2, dots, N_t$, $N_e$ is governed by the harmonic mean:
$$frac{1}{N_e} = frac{1}{t} sum_{i=1}^t frac{1}{N_i}$$
A single bottleneck generation severely depresses long-term genetic variation.
Figure 1.3: Wright-Fisher computer simulation of genetic drift demonstrating allele fixation/loss in small populations ($N = 10$) versus stability in large demes ($N = 500$), and Wright’s effective population size ($N_e$) equations.
🧪 Simulation Software & Physical Apparatus
- Physical Deme Modeling: Multi-well culture plates with 20 beads per well (10 diploid individuals) representing isolated island populations (founder populations).
- Computational Tool: R / Python script running 10,000 Wright-Fisher binomial draws:
rbinom(1, 2*N, p) / (2*N).
📋 Step-by-Step Simulation Protocol
- Small Deme Simulation ($N = 10$): Initialize 10 replicate populations with $N = 10$ individuals ($2N = 20$ alleles) at $p_0 = 0.50$. In each generation, draw 20 alleles with replacement according to current $p$. Track $p$ until every replicate either fixes ($p = 1.0$) or is lost ($p = 0.0$).
- Large Deme Comparison ($N = 500$): Repeat with $2N = 1000$ alleles over 50 generations. Observe the dampening of stochastic fluctuations.
- Founder Effect Simulation: From a large mainland pool ($p = 0.50$), draw 2 individuals ($N = 2$, 4 alleles) to colonize an empty island. Repeat 20 times. Record the dramatic disparity in initial island allele frequencies ($p = 0.0, 0.25, 0.50, 0.75, 1.0$).
- Effective Size Calculations: Compute $N_e$ for an endangered wildlife herd with 5 breeding males and 45 breeding females. Calculate $N_e$ for a population passing through sizes $1000 rightarrow 20 rightarrow 1000$ over 3 generations.
📊 Representative Simulation Results
Demographic Bottleneck Impact:
Generations: $N_1 = 1000$, $N_2 = 25$ (Catastrophic bottleneck), $N_3 = 1000$.
$$frac{1}{N_e} = frac{1}{3} left(frac{1}{1000} + frac{1}{25} + frac{1}{1000}right) = frac{1}{3} (0.001 + 0.040 + 0.001) = frac{0.042}{3} = 0.014$$
$$N_e = frac{1}{0.014} approx mathbf{71.4text{ individuals}}$$
Conservation Implication: Despite a census recovery back to 1000 animals in generation 3, the effective genetic population size is permanently reduced to ~71, causing enduring loss of heterozygosity and founder effect inbreeding depression.
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