Math 227 C : Introduction to Stochastic Differential Equations

Here we consider a new case study involving population genetics under the Wright-Fisher model. First we introduce a few jargons. Jargons 1.1 (Locus). A locus is simply a location on a chromosome. It can represent a gene, a SNP (single base pair polymorphism) or simply a location. Jargons 1.2 (Alleles). An allele is one of a number of alternative forms of the locus. A dominant allele is usually capitalized. For human and other diploid animals, there are typically two alleles of paternal and maternal origin. Jargons 1.3 (Genotype). A genotype is the genetic makeup of an individual, in this case, the types of alleles of an individual, e.g. Aa. Having the same pair of alleles is called homozygous while having different alleles is called heterozygous. Jargons 1.4 (Mutation). A mutation is a random genetic change. Here we refer to the change of one allele to another. Jargons 1.5 (Fitness). Fitness is the measure of survivability and ability to reproduce of an individual possessing a certain genotype. Jargons 1.6 (Neutral Evolution). Neutral evolution happens when all genotypes of interest have the same fitness. In this case, there is no selection from such genetic variations. Here we introduce the idea of Genetic Drift. Consider a model with the following assumptions: 1. There is a population with finite size N. 2. The size of the population is constant throughout evolution. 3. There are discrete, non-overlapping generations. 4. Mating is completely randomwith replacements. This means, an individual from generation i can give rise to, randomly, 0, 1 or more offsprings. From this, the Wright-Fisher model consider the following evolution along the generations: 1. At generation 0, there are 2N alleles some A, some a. 2. At generation 1, each A or a allele from generation 0 may result in one, more or zero copies of that allele. 3. Because the population size is finite and fixed, due to uneven passing of alleles by chance, eventually there will be only one allele, A or a left. This phenomenon is called genetic drift.