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Speaker: Saraí Hernández-Torres, Institute of Mathematics, National Autonomous University of Mexico
Title: Limit theorems for allele trees
Abstract: Mutations are a key driver of population change, giving rise to different versions of a gene known as alleles. As a result, a population’s genealogy is partitioned by genetic type, a structure encoded in an allele tree. Bertoin has studied limit theorems for allele trees under the assumption that each mutation introduces a novel allele, leading to an infinite number of possible types. However, research in computational biology has highlighted the need for models with a finite allele set.
In this talk, we introduce a model for the evolution of a population across discrete generations, where random mutations alter the genotype of individuals without affecting their offspring distribution. Whenever a mutation occurs, the new allele is chosen uniformly at random from a finite set of types, distinct from that of its parent. We then define the associated d-type allele tree and describe its scaling limit in the regime where the mutation rate is small and the total population size is critical. The talk is based on joint work with Airam Blancas and María Clara Fittipaldi.