Birth-Death Parameters

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Birth-Death Phylogenetic Trees

The birth-death process is a fundamental model in phylogenetics that generates random evolutionary trees. Species undergo speciation (birth) at rate λ and extinction (death) at rate μ. The resulting tree shape reveals diversification dynamics through time.

Key Concepts

Net diversification rate: r = λ - μ
When r > 0, lineages increase exponentially on average. When r ≤ 0, eventual extinction is certain.

Turnover ratio: ε = μ/λ
High turnover (ε → 1) means many species go extinct, creating asymmetric trees. Low turnover (ε → 0) is a pure-birth Yule process with balanced, symmetric trees.

Tree balance:
The shape of phylogenetic trees carries information about diversification. Symmetric trees (balanced) suggest constant-rate birth with little extinction. Asymmetric trees (unbalanced) suggest heterogeneous rates or high extinction.

Gillespie Algorithm

We simulate the birth-death process using the Gillespie algorithm for continuous-time stochastic processes:

  1. Start with 1 lineage at time t = 0
  2. With N active lineages, the total event rate is: λN + μN
  3. Time to next event: Δt = -ln(U) / (λN + μN), where U ~ Uniform(0,1)
  4. Event type: birth with probability λN/(λN + μN), otherwise death
  5. If birth: split a random lineage into two daughter lineages
  6. If death: remove a random lineage
  7. Continue until t ≥ T or all lineages extinct

Tree Statistics

Gamma (γ) statistic (Pybus & Harvey 2000):
Tests whether branching times follow a constant-rate birth-death model. γ < 0 suggests slowdown (early rapid diversification), γ > 0 suggests speedup (recent rapid diversification). Under constant rates, γ follows a standard normal distribution.

Lineages-Through-Time (LTT) plot:
The number of lineages N(t) vs time on a log scale. Under pure birth (Yule), this is linear with slope λ. Deviations reveal changes in diversification rate.

Model Variants

Yule process (μ = 0): Pure birth with exponential growth. Produces balanced trees with E[N(t)] = e^(λt).

Constant-rate birth-death: Both λ and μ constant. More realistic, allows extinction. Produces more asymmetric trees as μ/λ increases.

Trees are ultrametric: all tips (extant species) align at the present time, with branch lengths proportional to elapsed time.

Applications

  • Inferring speciation and extinction rates from molecular phylogenies
  • Testing hypotheses about adaptive radiation and key innovations
  • Understanding macroevolutionary dynamics and diversity patterns
  • Detecting shifts in diversification rates across clades