Chapter 10: Genetic Drift and Collapse Randomization = Stochastic Evolution

Not all evolution is adaptive. This chapter explores how ψ = ψ(ψ) incorporates randomness through genetic drift, creating evolutionary trajectories independent of fitness.

10.1 The Drift Function

Definition 10.1 (Genetic Drift): Random sampling of alleles across generations: $\text{Var}(p_{t+1}) = \frac{p_t(1-p_t)}{2N_e}$

where $p$ is allele frequency and $N_e$ is effective population size.

Drift causes:

• Random frequency changes
• Eventual fixation or loss
• Divergence without selection
• Reduced variation

10.2 The Wright-Fisher Model

Theorem 10.1 (Sampling Dynamics): In finite populations: $P(k|n,p) = \binom{2N}{k} p^k (1-p)^{2N-k}$

where $k$ alleles are sampled from $2N$ gene copies.

Properties:

• Binomial sampling each generation
• Variance inversely proportional to size
• No directional bias
• Markov process

Proof: Random mating with replacement creates binomial sampling of gametes. ∎

10.3 Effective Population Size

Real populations deviate from ideal:

$N_e = \frac{4N_m N_f}{N_m + N_f}$

for unequal sex ratios.

Factors reducing $N_e$:

• Non-random mating
• Varying reproductive success
• Age structure
• Non-constant population size

$N_e \ll N_{\text{census}}$ typically.

10.4 Fixation Probability

Definition 10.2 (Neutral Fixation): Probability equals initial frequency: $P_{\text{fix}} = p_0$

For new mutations: $P_{\text{fix}} = \frac{1}{2N}$

Time to fixation: $\bar{t}_{\text{fix}} = 4N_e \text{ generations}$

Selection modifies these neutrality expectations.

10.5 Nearly Neutral Theory

Theorem 10.2 (Selection-Drift Balance): Selection effectiveness depends on $N_e$: $|2N_e s| < 1 \Rightarrow \text{Effectively neutral}$

where $s$ is selection coefficient.

Implications:

• Small populations: More drift
• Large populations: More selection
• Slightly deleterious mutations can fix
• Molecular evolution rate varies

10.6 Bottlenecks and Founder Effects

Population crashes amplify drift:

$N_e^{\text{harmonic}} = \frac{t}{\sum_{i=1}^t \frac{1}{N_i}}$

Bottleneck consequences:

• Rapid allele frequency changes
• Loss of rare alleles
• Increased homozygosity
• Reduced adaptive potential

Shaping species' evolutionary trajectories.

10.7 Genetic Draft

Definition 10.3 (Hitchhiking): Linked selection affects neutral sites: $\text{Diversity} = \theta \cdot \exp(-\rho/c)$

where $\rho$ is recombination rate and $c$ is selection strength.

Selective sweeps create:

• Reduced variation near selected sites
• Linkage disequilibrium
• Apparent molecular clocks
• Drafting of neutral alleles

10.8 Coalescent Theory

Tracing lineages backward:

$P(\text{coalescence}) = \frac{1}{2N_e}$

per generation for two lineages.

Applications:

• Estimating population history
• Dating common ancestors
• Detecting selection
• Phylogeography

Time runs backward in coalescent models.

10.9 Drift in Structured Populations

Theorem 10.3 (Spatial Drift): Subdivision increases differentiation: $F_{ST} = \frac{1}{1 + 4N_e m}$

where $m$ is migration rate.

Creating:

• Local differentiation
• Wahlund effect
• Isolation by distance
• Metapopulation dynamics

10.10 Molecular Clocks

Drift drives constant evolution:

$K = 2\mu$

where $K$ is substitution rate and $\mu$ is mutation rate.

Clock properties:

• Independent of population size
• Approximately constant rate
• Useful for dating
• Violated by selection

10.11 Drift vs Selection

The eternal balance:

$\text{Evolution} = \text{Selection} + \text{Drift} + \text{Interaction}$

Selection dominates when: $|s| > 1/(2N_e)$ Drift dominates when: $|s| < 1/(2N_e)$ Both important when: $|s| \approx 1/(2N_e)$

Creating complex dynamics.

10.12 The Drift Paradox

Random changes seem wasteful, yet drift is universal:

Apparent waste: Most drift changes are non-adaptive Hidden value: Exploration of neutral space

Resolution: Drift provides evolution's random search algorithm, exploring regions of genotype space that selection alone would never reach. This stochastic wandering discovers neutral networks—sets of genotypes with equivalent fitness but different mutational neighborhoods. When environments change, yesterday's neutral variant may become today's adaptation. Drift ensures that ψ maintains exploratory breadth even when selection narrows focus. In the interplay of chance and necessity, evolution finds both immediate optimization and long-term flexibility.

The Tenth Echo

Genetic drift reveals ψ's incorporation of chance into the evolutionary algorithm. Like Brownian motion at the molecular level, drift creates random walks through genetic space, ensuring that evolution explores broadly rather than getting trapped on local fitness peaks. This stochastic component seems to oppose the directionality of selection, yet both work together—selection providing direction, drift providing exploration. In embracing randomness, ψ ensures that evolution remains creative, discovering solutions that pure optimization would miss.

Next: Chapter 11 examines ψ-Potential in Adaptive Landscapes, exploring how fitness topographies shape evolutionary possibilities.