Group: Chase Stolworthy

Title: Genetic Drift Modeled by a Markov Chain

Abstract: Genetic drift is an evolutionary process responsible for the
change in frequency of alleles in a population due to the random sampling
of organisms. The Wright-Fisher model of genetic drift can be modeled as a
discrete-time Markov chain. My project will explore how this model works,
use topics of linear algebra to show interesting properties of the model,
and run computer simulations of the Markov chain on populations of
different sizes with varying allele frequency distributions.

Outline:
- Genetic Drift
- Defined a the change in allele frequencies due to random sampling of
organisms
- Sources of randomness from independent assortment of chromosomes and
demographic stochasticity
- Therefore, genetic drift is always happening
- Wright-Fisher Model
- Consider a population of constant size N and gene with variants, A1 and A2
- Assume organisms are diploid, then there are 2N alleles (gene copies) in
population
- The mating process happens all at once in discrete time intervals
- Create the next generation by choosing 2N alleles with replacement from
current generation
- Markov Chain
- The Wright-Fisher model is a discrete-time Markov chain, that is, the
allele frequencies of the next generation only depend on the frequencies of
the current generation
- For the probability vector, each entry x_i, 0 <= i <= 2N, is the the
probability of i alleles of type A1 in the population
- If there are i alleles of type A1 in the current generation, then the
number of A alleles in the next generation is a binomial random variable
with n=2N and p=i/2N
- Then, the stochastic matrix is given by p_ij = (2N choose
j)(i/2N)^j(1-i/2N)^(2N-j)
- Note that p_0,0 = 1 and P_2N,2N = 1, because at those states the A1
allele has become fixed in the population