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Mathematical Concepts in Genetics: Inheritance, Diagonalization, and Probabilities

This resource covers essential mathematical concepts applied to genetics, including eigenvalues, eigenvectors, and matrix diagonalization. It emphasizes intuitive understanding of limits and inherits probabilities for genotypes. The focus is on autosomal and X-linked inheritance patterns, using examples to demonstrate genotype distributions in plant populations and inbreeding scenarios. Readers will gain valuable insights into calculating genetic probabilities, enhancing their comprehension of how mathematics intersects with genetic principles.

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Mathematical Concepts in Genetics: Inheritance, Diagonalization, and Probabilities

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  1. Genetics Anton 10.16

  2. Mathematical concepts • Eigenvalues and Eigenvectors • Diagonalization of a Matrix • Intuitive Understanding of Limits • Probability

  3. Genotypes

  4. Genotypes A = dominant trait a = recessive trait

  5. Genotypes A a

  6. Genotypes A = brown eyes a= blue eyes AA = Aa= aa = brown eyes brown eyes blue eyes

  7. Probability of Inheritance

  8. Probability of Inheritance

  9. Example: Autosomal Inheritance • A population of plants consists of some distribution of genotypes AA, Aa, and aa. • Each plant is fertilized with another plant of genotype AA. • What is the distribution of genotypes AA, Aa, and aa after n generations?

  10. Example: Autosomal Inheritance • Each plant is fertilized with another plant of genotype AA.

  11. Example: Autosomal Inheritance AA: Aa: aa:

  12. Example: Autosomal Inheritance where and

  13. Example: Autosomal Inheritance

  14. Example: Autosomal Inheritance Can you calculate ? Can you calculate ?

  15. Example: Autosomal Inheritance • If prove that .

  16. Example: Autosomal Inheritance Diagonalization of matrix Eigenvalues: Eigenvectors:

  17. Example: Autosomal Inheritance Eigenvalues: Eigenvectors:

  18. Example: Autosomal Inheritance

  19. Example: Autosomal Inheritance

  20. Example: Autosomal Inheritance

  21. X-linked Inheritance Carries gene Does not carry gene

  22. Probability of X-linked Inheritance A A A a A AA AA a Aa aa

  23. Probability of X-linked Inheritance A a A a A A A A

  24. Probability of X-linked Inheritance

  25. Example 2: X-linked inheritance In-breeding: • Begin with a male and female. • Select two of their offspring, one male and one female, and mate them. • Select two of the resulting offspring, one male and one female, and mate them. • Etc.

  26. Example 2: X-linked Inheritance

  27. Example 2: X-linked Inheritance

  28. Example 2: X-linked Inheritance

  29. Example 2: X-linked Inheritance

  30. Example 2: X-linked Inheritance Solve where

  31. Example 2: X-linked Inheritance Solve where

  32. Example 2: X-linked Inheritance

  33. Example 2: X-lined Inheritance

  34. Example 2: X-linked Inheritance

  35. Example 2: X-linked Inheritance Initial Parents , i.e. , then

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