Experiment 02Natural SelectionFitness Dynamics
Simulating Microevolutionary Forces: Natural Selection against Recessive Lethals vs Heterozygote Advantage (Sickle-Cell Model)
Duration: 4 Hours • Monte Carlo / Population Bench Model
🎯 Experimental Aim & Learning Objectives
To experimentally simulate multi-generational allele frequency trajectory shifts under directional selection against a completely recessive lethal allele ($s = 1.0$), demonstrate the declining efficiency of natural selection as lethal alleles become rare, and contrast this with overdominant balancing selection (heterozygote advantage) maintaining stable polymorphism (the sickle-cell hemoglobin $HbS$ malaria resistance paradigm).
🔬 Theoretical Foundation & Mathematical Selection Models
Natural selection acts as a deterministic evolutionary force altering allele frequencies based on differential reproductive success (relative fitness, $w$):
- Selection Against Recessive Lethal ($w_{AA} = 1, w_{Aa} = 1, w_{aa} = 0$): All homozygous recessive zygotes die prior to reproductive maturity ($s = 1.0$). The recessive allele frequency $q_t$ after $t$ generations decays as:
$$q_t = frac{q_0}{1 + t cdot q_0}$$
The number of generations required to reduce the frequency from $q_0$ to $q_t$ is: $t = frac{1}{q_t} – frac{1}{q_0}$. As $q$ becomes very small, almost all recessive alleles reside in heterozygous carriers ($Aa$), making complete elimination by natural selection virtually impossible. - Overdominant Balancing Selection (Heterozygote Advantage): When the heterozygote exhibits higher fitness than either homozygote ($w_{Aa} > w_{AA}$ and $w_{Aa} > w_{aa}$):
• $w_{AA} = 1 – s_1$, $w_{Aa} = 1$, $w_{aa} = 1 – s_2$
The population converges onto a stable internal equilibrium allele frequency ($hat{q}$):
$$hat{q} = frac{s_1}{s_1 + s_2}, quad hat{p} = frac{s_2}{s_1 + s_2}$$
In Sub-Saharan Africa, where $HbA/HbA$ individuals suffer mortality from Plasmodium falciparum malaria ($s_1 approx 0.12$) and $HbS/HbS$ individuals suffer severe sickle-cell anemia ($s_2 approx 0.85$), balancing selection actively maintains the lethal $HbS$ allele at $hat{q} approx 0.124$ (12.4%).
Figure 1.2: Purging kinetics of a recessive lethal allele under directional selection ($s = 1.0$) compared with overdominant balancing selection maintaining the sickle-cell allele ($HbS$).
🧪 Simulation Materials & Protocol
- Benchtop Simulation Kit: Large sorting containers with 500 red beads (allele $A$) and 500 green beads (allele $a$). Genotype collection cups labeled $AA$, $Aa$, and $aa$ (lethal discard bin).
- Digital Simulator: Populus 5.5 / R script running deterministic recurrence equations.
📋 Step-by-Step Simulation Protocol
- Generation 0 Initialization: Mix 500 red ($A$) and 500 green ($a$) beads. $p_0 = q_0 = 0.50$. Draw 100 random pairs to form 100 individuals.
- Recessive Lethal Selection Step: Completely discard all $aa$ pairs (100% mortality, $s = 1$). Retain all $AA$ and $Aa$ pairs.
- Gene Pool Replenishment: Count surviving alleles in $AA$ and $Aa$ survivors. Calculate new $q_1$. Replenish the bead container to maintain total pool proportion. Repeat for generations $t = 1, 2, 3, 4, 5$.
- Overdominance Simulation: Reset pool. Impose selective penalties: discard a fraction $s_1 = 0.20$ of $AA$ pairs and $s_2 = 0.80$ of $aa$ pairs. Keep 100% of $Aa$ heterozygotes. Recompute frequencies across 10 cycles until $hat{q}$ stabilizes.
📊 Representative Multigenerational Trajectory Data
| Generation ($t$) | Recessive Lethal $q_t$ (Theoretical) | Recessive Lethal $q_t$ (Simulated) | Balancing Selection $q_t$ ($s_1=0.2, s_2=0.8$) |
|---|---|---|---|
| 0 | 0.500 | 0.500 | 0.500 |
| 1 | 0.333 | 0.328 | 0.320 |
| 2 | 0.250 | 0.245 | 0.245 |
| 5 | 0.143 | 0.138 | 0.202 (Equilibrium → 0.200) |
| 10 | 0.083 | 0.080 | 0.200 (Stable $hat{q}$) |
Evolutionary Conclusion: Purging efficiency plummets as $q$ decreases because $99%$ of lethal alleles reside in phenotypically normal heterozygotes. In contrast, overdominance drives the population inexorably to the stable attractor point $hat{q} = frac{0.2}{0.2 + 0.8} = 0.200$, maintaining genetic diversity permanently.
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