Coalescent Processes: Reversed Branching

Coalescence

Branching viewed backward is coalescence, the process of merging or clumping. It arises naturally in the structuring or formation of dispersed matter of various kinds and in various scales, from that of molecule aggregates in colloids (so-called micelles) to galaxies. It has been studied by physicists through computer simulation (e.g., Nilsson et al. 2000) and in a series of interesting mathematical articles by Aldous (1999).

Evolution can be viewed as a grand multi-type branching process, with new species that arise through mutation (see Jagers 1991; Jagers et al. 1992; Taib 1992). The study of the origin of species is then time-reversed branching (i.e., coalescence). In genetics, the latter is also used to trace the roots of the genetic composition of populations and its development. It was within this area that the first pure coalescence model, the Kingman coalescent (1982a), was formulated as a reverse counterpart to the diffusion approximation of the renowned Wright–Fisher model (Fisher 1930; Wright 1931; see also Ewens 1979).

The object of genetic models is thus population composition rather than size. Indeed, most population genetics even assumes that population size is completely constant over generations. As we show later, the Wright–Fisher model can be obtained as Galton–Watson branching with Poisson reproduction, conditioned at a constant population size. In the same vein, most population genetics simplifies the flow of time into generation counting. Instead, it is lineage that counts. What are the relations among n individuals sampled …