THE NUMBER OF GENERATIONS BETWEEN BRANCHING EVENTS IN A GALTON–WATSON TREE AND ITS APPLICATION TO HUMAN MITOCHONDRIAL DNA EVOLUTION

We have shown that the problem of existence of a mitochondrial Eve can be understood as an application of the Galton–Watson branching process and presents interesting analogies with critical phenomena in Statistical Mechanics. We shall review some of these results here. In order to consider mutations in the Galton–Watson framework, we shall derive a general formula for the number of generations between successive branching events in a pruned genealogic tree. We show that in the supercritical regime of the Galton–Watson model, this number of generations is a random variable with geometric distribution. In the critical regime, population in the Galton–Watson model is of constant size in the average. Serva worked on genealogic distances in a model of haploid constant population and discovered that genealogic distances between individuals fluctuated wildly both in time and in the realization of the model. Also, such fluctuations seem not to disappear when the population size tends to infinity. This phenomenon was termed lack of self-averaging in the genealogic distances. Although in our model, population is not strictly constant, we argue that the lack of self-averaging in the genealogic distance between individuals may be viewed as a consequence of being exactly at or near a critical point.