Extinction Probability of Some Recessive Alleles of X-Linked Genes in the Context of Two-Sex Branching Processes

Abstract

Some defective alleles of certain genes can cause severe diseases or serious disorders in the organisms that carry them. Some of these genes, as could be the responsible for hemophilia, are linked to X chromosome. If the alleles causing the disorder are dominant, all the carriers are affected and most of them do not reach breeding age so they are rarely detected in a population. However, recessive pernicious alleles can survive since they only affect to carrier males and homozygous carrier females (the last ones must be daughters of a carrier male, so they rarely exist). Hence, heterozygous carrier females are not affected but can pass the allele onto offspring. In this work, we introduce a multitype two-sex branching process for describing the evolution of the number of individuals carrying the alleles, R and r, of a gene linked to X chromosome. The R allele is considered dominant and the r allele is supposed to be recessive and defective, responsible of a disorder. In this model we study the fate of the recessive allele in the population, i.e. under which conditions it eventually disappears. We also investigate conditions for the fixation of the dominant allele and the extinction of the population. Finally, conditions for the coexistence of both alleles are conjectured and their proofs are proposed as open problems.