Coalescent Gene Genealogies

The Coalescent Process

The coalescent is a backward-time view of population genetics. Instead of tracking alleles forward through generations (as in Wright-Fisher models), we trace sampled lineages backward to their most recent common ancestor (MRCA).

Kingman's Coalescent

In a large population of size N, when we trace k lineages backward in time, the probability that two lineages coalesce in the previous generation is approximately k(k-1)/(4N). The time until the next coalescent event follows an exponential distribution with rate k(k-1)/(4N).

• Time to MRCA: The total time to reach the most recent common ancestor depends on the number of sampled lineages and population size. For n samples, the expected time is approximately 2N(1 - 1/n) generations.
• Tree shape: Most coalescent events occur near the tips (present day), with long branches near the root. This creates characteristically imbalanced genealogies.
• Population size effect: Larger populations have longer coalescent times. The coalescent time scale is measured in units of 2N generations.
• Structured coalescent: With migration between populations, lineages can jump between demes as we trace them backward, affecting both tree shape and coalescent times.

Wright-Fisher Reference

The Wright-Fisher model simulates allele frequencies forward in time through discrete generations with random mating. The coalescent emerges as the limiting genealogical process when N is large. Enable the Wright-Fisher comparison to see how forward-time sampling produces similar genealogical structures.

Key Insights

• Coalescent theory provides computational efficiency: instead of simulating entire populations, we only track the genealogy of sampled individuals.
• The exponential waiting times between coalescent events create highly variable tree shapes.
• Recent history (near the tips) contains more coalescent events than ancient history (near the root).
• Migration creates a more complex pattern where lineages can trace back through multiple populations.