LinearAlgebraandGenetics - Linear Algebra and Genetics 1...

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Linear Algebra and Genetics 1 Linear Algebra and Genetics Caroline Carr Math 340, Advanced Linear Algebra Professor Debnath May 3, 2006
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Linear Algebra and Genetics 2 Abstract This paper will discuss how linear algebra can be applied to genetic sciences. Topics discussed will include autosomal inheritance, x-linked inheritance, and inheritance trends of multiple alleles. It will be demonstrated that linear algebra can not only predict the outcome of first generation offspring, but also how to predict the long- term presence of certain genetic traits. A basic understand of linear algebra would be beneficial, but not entirely necessary.
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Linear Algebra and Genetics 3 Table of Contents I. Autosomal Inheritance Probabilities II. Autosomal Inheritance and Genetic Recessive Diseases III. X-Linked Inheritance IV. Markov Chains and Genetics V. Linear Algebra and the GPI
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Linear Algebra and Genetics 4 Over the years, the study of genetics has become increasingly more important. Not only do parents want to know what their children will look like and whether or not they will be healthy, but medical professionals also wonder if certain diseases will continue to affect future generation. Linear algebra can be used to answer all of these questions. I. Autosomal Inheritance Probabilities The most basic inheritance is the autosomal inheritance, which controls many physical characteristics of an plants and humans, including plant or eye color. In autosomal inheritance, each of the two parents have two genes that comprise his or her genotype (Rorres 1977). The possible genes are denoted A , the dominant gene, and a , the recessive gene. In this type of inheritance, the possible genotypes are AA, Aa, or aa . An AA genotype would demonstrate the dominant trait; an Aa genotype would exhibit either a blended trait (if A corresponded to the color red and a , white, the offspring would be pink) or the dominant trait; while an aa genotype would correspond to the recessive trait. If the genotype of each adult is known, the probability of the offspring’s genotype can be easily determined. Each adult will pass on one of its two genes to the offspring, making four possible combinations for the new genotype. If an AA adult mated with an Aa adult, the offspring could either be AA, AA, Aa, or Aa , thus the probability that an offspring will be AA is ½ , and the same for Aa . There would be a zero probability that an offspring would have the genotype aa. The following table models a typical punnett square, which is used to determine the possible genotypes of an offspring: Parent1 Gene1 (P1G1) Parent1 Gene2 (P1G2) Parent2 Gene1 (P2G1) P1G1,P2G1 P1G2,P2G1 Parent2 Gene2 (P2G2) P1G1,P2G2 P1G1,P2G2
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Linear Algebra and Genetics 5
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