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Unformatted text preview: Linear Algebra in Genetics Carter Butaud Michael Yamamoto June 5, 2008 1 1 Background: Biologists often study the genetic makeup of a plant or animal in order to make predictions about its descendants. In the study of autosomal inheritance, each characteristic studied is taken to be the result of two genes. For example, suppose we were to look at eye color (only considering blue and brown eyes for now). We could represent the gene for blue eyes as a and for brown eyes as A. A person could have three different combinations of these genes: AA, Aa, or aa. A plant or animals genotype is all of its gene combinations. In any set of genes, one gene is said to be dominant (and we generally denote that gene with the capital letter). So in the example given above, both combinations AA and Aa would produce an individual with brown eyes. The other gene (in our example, the blue- eyed gene) is said to be recessive. When a plant or animal inherits genes from its parents, each gene pair is made up of the two genes from its parents, with equal probability of having either gene. So a child with one AA parent and one aa parent could be any of the three combinations, while the descendants of an Aa and an aa could be either Aa or aa. It is possible to represent the possible descendants of any two parents with a matrix. If we look at a distribution of different gene types to start out with, we can model the probable makeup of the subsequent generations by raising that matrix to a power and then multiplying by the original makeup. Diagonalizing that matrix makes it easy to compute the probable makeup of an arbitrary generation of descendants and, more importantly, to find the limit as those generations approach infinity of the distribution of genes if it exists. 2 Problems: In autosomal inheritance, offspring inherit one gene of the pair belonging to each parent in order to get a pair. For the following, assume that either of a parents genes is equally likely to be inherited. 1. A farmer has a large population of plants consisting of some distribution of all three possible genotypes: AA, Aa, aa. First show that the following table correctly describes the probabilities of the possible genotypes of the offspring for all possible combinations of the genotypes of the parents....
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