03/20/2024
By Jong Soo Lee

The Kennedy College of Sciences, Department of Mathematics & Statistics, invites you to attend a Master’s thesis defense by Rebecca Mendum on “Mathematical Tools for Population Genetics."

Candidate Name: Rebecca Mendum
Degree: Master’s (MS)
Defense Date: Wednesday, April 3, 2024
Time: 10:30 to 11:30 a.m.
Location: Pulichino-Tong (PTB) Room 224, North Campus

Thesis Title: Mathematical Tools for Population Genetics

Committee:

  • Advisor Jong Soo Lee, PhD, Department of Mathematics & Statistics, UMass Lowell
  • Daniel Klain, PhD, Department of Mathematics & Statistics, UMass Lowell
  • Aida Kadic-Galeb, PhD, Department of Mathematics & Statistics, UMass Lowell

Brief Abstract:
In this work, we discuss several population genetics models and investigate their limiting behavior using Markov chains. In the classical Hardy-Weinberg model, gene frequencies are constant over time, which can be easily derived and computed. The sib-mating model extends the Hardy-Weinberg model and can allow for mutation and selection; both models are in an infinite population. The neutral (without mutation) Wright-Fisher Model uses a finite population and a binomial random variable to illustrate changes in the proportion of A alleles. Finally, we use the Moran model, which is a birth-and-death chain and can handle overlapping generations, unlike the Hardy-Weinberg and Wright-Fisher models.

We find that during a process called random genetic drift, populations move toward states including only one type of allele A or a, where all individuals are homozygous for a gene at a specific locus. We compute long-term limiting probabilities for a variety of initial probability distributions and population sizes. Furthermore, we compute absorbing probabilities in systems where there are absorbing states in the model. Then, we introduce mutations to the Wright-Fisher and Moran models, in which there are no longer absorbing states. We investigate several rates of mutation in a two-way mutation process. Throughout, we derive mathematical results and provide computational evidence for our results. We confirm and expand on results from other sources.