Bisexual branching processes to model extinction conditions for Y-linked genes

Abstract

In a two-sex monogamic population, the evolution of the number of carriers of the two alleles of a Y-linked gene is considered. To this end, a multitype bisexual branching model is presented in which it is assumed that the gene has no influence on the mating process. It is deduced from this model that the average numbers of female and male descendants per mating unit constitute the key to determining the extinction or survival of each allele. Moreover, the destiny of each allele in the population is found not to depend on the behaviour of the other.

Limiting genotype frequencies of Y-linked genes through bisexual branching processes with blind choice

“…This statement was proved in González et al (2009) (see Result 6 therein) if the probability α for an offspring to be female is different from 0.5. The case α = 0.5 is included in the following result.…”

“…The following model, introduced by González et al (2009), describes the evolution of the number of carriers of a Y-linked gene in a two-sex monogamous population. The gene occurs in two allelic forms, denoted R and r. Since the Y-chromosome is haploid and specific to males, the population is formed by females and by two types of male, denoted R-and r-males, depending on which allele they carry.…”

“…It is shown in González et al (2009) that each genotype shows the dual behaviour typical for branching processes and known as the extinction-explosion dichotomy. This means that the number of couples of any type is bound to undergo either extinction or indefinite growth.…”

“…3 Survival of only one genotype: Limiting growth rate A necessary and sufficient condition for a genotype to have positive probability of fixation is that both the female and the male mean offspring per couple of that genotype are greater than unity (see Result 2 in González et al (2009)). This is due to the fact that, if fixation of a particular allele has occurred, the corresponding genotype evolves essentially as a bisexual branching process with perfect fidelity mating and the reproduction law of the surviving genotype.…”

“…Roughly speaking, these processes form an extension of classical two-type Galton-Watson branching processes by additionally imposing a mating structure. González et al (2009) have recently introduced a model of this kind for the evolution of Y-linked genes which occur in two allelic forms, called R and r. They assume monogamous mating (mating with perfect fidelity) with blind choice, which means that females choose their mate without recognizing or caring about his genotype. The latter condition may be justified by the fact that Y-linked genes are typically not expressed in males, or, if they are, do not have any preferential impact on the mating process.…”