The multi-type Galton-Watson process in a genetical context

In 1966 Ewens noted that the probability of fixation of a favourable new mutant linked to another gene is practically independent of the population size for a large random-mating population, and that what normally determines the fate of a new mutant gene is the behaviour of its frequency while it is rare. Because the relative frequency of the new mutant is small initially, he was able to obtain a linear recurrence relation over time for the expected relative frequencies of the two gametes which contain the mutant. This is a very interesting result, and immediately suggests a two-type Galton-Watson process as a suitable discrete-time stochastic model. In the present paper, some techniques associated with the multi-type Galton-Watson process and developed in the field of population mathematics are used to study the frequency of occurrence of a rare mutant gene. The use of these techniques in practice is demonstrated by the solution of a numerical example.