Chapter 21: ψ-Rates of Evolutionary Change = Tempo and Mode

Evolution's clock ticks at different speeds. This chapter examines how ψ = ψ(ψ) manifests across temporal scales, from molecular substitutions to morphological transformations.

21.1 The Rate Function

Definition 21.1 (Evolutionary Rate): Change per unit time: $r = \frac{\Delta \psi}{\Delta t}$

Measurable at multiple levels:

• Molecular (substitutions/site/year)
• Morphological (darwins)
• Taxonomic (lineages/million years)
• Functional (innovations/epoch)

21.2 Molecular Clock Hypothesis

Theorem 21.1 (Constant Substitution): Neutral mutations accumulate steadily: $K = 2N\mu \times \frac{1}{2N} = \mu$

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

Proof: Neutral mutations fix with probability 1/2N, arising at rate 2Nμ. ∎

Creating time-proportional divergence.

21.3 Rate Heterogeneity

Clocks tick differently:

$r_i = r_0 \times f(\text{Generation time}, \text{Population size}, \text{Selection})$

Sources of variation:

• Metabolic rate effects
• DNA repair efficiency
• Effective population size
• Selection intensity
• Environmental mutagenesis

21.4 The Darwin Unit

Definition 21.2 (Morphological Rate): Proportional change: $d = \frac{\ln(x_2) - \ln(x_1)}{\Delta t} \times 10^6$

measured in darwins (d).

Typical rates:

• Laboratory selection: 10,000-60,000 d
• Post-glacial evolution: 1,000-10,000 d
• Fossil record: 0.1-10 d
• Living fossils: ~0 d

21.5 Punctuated Equilibrium

Theorem 21.2 (Stasis and Bursts): Evolution concentrated in speciation: $\text{Rate}_{\text{anagenesis}} \ll \text{Rate}_{\text{cladogenesis}}$

Pattern:

1. Long stasis (morphological equilibrium)
2. Rapid change (during speciation)
3. New stasis (daughter species)

Challenging gradualism.

21.6 Adaptive Radiation Rates

Explosive diversification:

$\frac{dS}{dt} = \lambda_0 \exp(-\alpha t)$

where initial speciation rate $\lambda_0$ declines as niches fill.

Examples:

• Cambrian explosion: Body plans in 20 MY
• Hawaiian silverswords: 30 species in 5 MY
• African lake cichlids: 500+ species in 15,000 years

21.7 Living Fossils

Definition 21.3 (Evolutionary Stasis): Minimal change over deep time: $|\psi(t) - \psi(0)| < \epsilon \text{ for } t > 10^8 \text{ years}$

Examples:

• Coelacanths (400 MY)
• Horseshoe crabs (450 MY)
• Ginkgo trees (270 MY)
• Stromatolites (3.5 BY)

Stability through environmental constancy or optimized design.

21.8 Rapid Evolution Examples

Theorem 21.3 (Contemporary Evolution): Observable within decades: $\Delta\bar{z} = h^2 \times S$

where response to selection is immediate.

Documented cases:

• Darwin's finches (beak size)
• Peppered moths (melanism)
• Guppies (predation response)
• Anolis lizards (island colonization)
• Antibiotic resistance (days)

21.9 Developmental System Drift

Hidden evolution:

$\text{Phenotype}(t) = \text{constant}$ $\text{Development}(t) \neq \text{Development}(0)$

Developmental drift: Different genetic routes to same phenotype Genetic redundancy: Multiple solutions to same problem Cryptic variation: Hidden potential revealed by perturbation

21.10 Extinction Rates

Definition 21.4 (Lineage Termination): $\mu = -\frac{\ln(S_t/S_0)}{t}$

Background vs mass extinction:

• Background: 0.1-1 extinctions/MY
• Mass extinction: >75% species loss
• Current: 100-1000× background

Extinction shapes opportunity.

21.11 Coevolutionary Rates

Coupled evolution:

$\frac{d\psi_1}{dt} = f(\psi_2)$ $\frac{d\psi_2}{dt} = g(\psi_1)$

Red Queen dynamics: Constant evolution to maintain fitness Arms races: Escalating adaptations Mutualism evolution: Coordinated change

Rates linked across species.

21.12 The Rate Paradox

Evolution is simultaneously too fast and too slow:

Too fast: Laboratory rates would transform species in centuries Too slow: Stasis persists for millions of years Variable: Same lineage shows both patterns

Resolution: Evolutionary rates reflect the changing relationship between organisms and environments. During stable periods, stabilizing selection maintains optimal phenotypes, creating apparent stasis despite ongoing molecular evolution. When environments shift or new opportunities arise, directional selection can drive rapid change. The rate of evolution is thus not a constant but a variable that tracks the degree of adaptive mismatch. ψ evolves quickly when far from optima, slowly when well-adapted. This creates a punctuated pattern where long periods of fine-tuning alternate with bursts of innovation. Time itself is thus relativistic in evolution—measured not by the clock but by the pressure to change.

The Twenty-First Echo

Evolutionary rates reveal time's plasticity in ψ's domain. What seems impossibly slow in human terms—the gradual modification of a beak or the shift of a coastline—can be blindingly fast in geological time. Conversely, what appears frozen in the fossil record may hide furious molecular evolution maintaining the phenotypic status quo. By measuring evolution's varied tempos, we learn that ψ is neither inherently conservative nor revolutionary but responsive—changing at the pace demanded by circumstance. In this temporal flexibility lies evolution's power to both preserve and transform.

Next: Chapter 22 explores Gradualism vs Punctuated ψ-Equilibrium, examining patterns of evolutionary change.