Abstract
A multidimensional two-sex branching process is introduced to model the evolution of a pedigree originating from the mutation of an allele of a Y-linked gene in a monogamous population. The study of the extinction of the mutant allele and the analysis of the dominant allele in the pedigree is addressed on the basis of the classical theory of multi-type branching processes. The asymptotic behavior of the number of couples of different types in the pedigree is also derived. Finally, using the estimates of the mean growth rates of the allele and its mutation provided by a Gibbs sampler, a real Y-linked pedigree associated with hearing loss is analyzed, concluding that this mutation will persist in the population although without dominating the pedigree.
Highlights
Branching Processes (BPs) are stochastic mathematical models widely used in population genetics as an alternative approach to the Wright–Fisher model when the classical assumption of constant population size is dropped
We shall introduce a discrete-time Y-linked two-sex BP, which determines generation-by-generation the evolution of a pedigree associated with a Y-linked gene with two alleles, r and R, where R originates from a mutation of the r allele and will be the target characteristic
It indicates that the evolution of the observed Y-linked pedigree was in agreement with the behavior predicted by the two-sex BPs introduced in this communication
Summary
Branching Processes (BPs) (see [1]) are stochastic mathematical models widely used in population genetics (see [2]) as an alternative approach to the Wright–Fisher model when the classical assumption of constant population size is dropped (see [3]). Mathematics 2020, 8, 256 couple formed in a generation depends on the total number of females and males in that generation and is determined by the mating scheme (preference or blind choice) that defines the sexual interaction Those studies considered an overall population, dealing with such problems as its extinction, the fixation of specific alleles, the coexistence of different alleles, their rates of growth, etc. These features are shown, which illustrates the evolution of the first generations of a simulated Y-linked pedigree evolving within an overall population Notice that, in this genetic framework (the Y-linked gene case), the evolution of the number of carriers of different genotypes in the overall population could be modeled by the Y-linked two-sex BP with the mutant allele that was introduced in [16].
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