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Genetics
Anton 10.16
Mathematical concepts
• Eigenvalues and Eigenvectors• Diagonalization of a Matrix• Intuitive Understanding of Limits
• Probability
Genotypes
Genotypes
A = dominant trait
a = recessive trait
Genotypes
Aa
Genotypes
A = brown eyes
a = blue eyes
AA =
Aa =
aa =
brown eyes
brown eyes
blue eyes
Probability of Inheritance
Genotype ProbabilityAA ¼ Aa ½ aa ¼
Probability of Inheritance
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?
Example: Autosomal Inheritance
• Each plant is fertilized with another plant of genotype AA.
Example: Autosomal Inheritance
AA:
Aa:
aa:
Example: Autosomal Inheritance
where and
Example: Autosomal Inheritance
Example: Autosomal Inheritance
Can you calculate ?Can you calculate ?
Example: Autosomal Inheritance
• If prove that .
Example: Autosomal Inheritance
Diagonalization of matrix Eigenvalues: Eigenvectors:
Example: Autosomal Inheritance
Eigenvalues:
Eigenvectors:
Example: Autosomal Inheritance
𝐷𝑛=[1 0 0
0120
0 0 0]𝑛
=[1𝑛 0 0
0 ( 12 )𝑛
0
0 0 0𝑛]=[ 1 0 0
0 ( 12 )𝑛
0
0 0 0]
Example: Autosomal Inheritance
1
Example: Autosomal Inheritance
X-linked Inheritance
Carries gene Does not
carry gene
Probability of X-linked Inheritance
A AA AA AA
A aa Aa aa
Probability of X-linked Inheritance
A aA a
A AA A
Probability of X-linked Inheritance
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.
Example 2: X-linked Inheritance
Parents
(A, AA)
Offspring
(A, AA) 1
(A, Aa) 0
(A, aa) 0
(a, AA) 0
(a, Aa) 0
(a, aa) 0
Example 2: X-linked InheritanceParents
(A, AA) (A, Aa)
Offspring
(A, AA) 1 1/4
(A, Aa) 0 1/4
(A, aa) 0 0
(a, AA) 0 1/4
(a, Aa) 0 1/4
(a, aa) 0 0
Example 2: X-linked InheritanceParents
(A, AA) (A, Aa) (A, aa)
Offspring
(A, AA) 1 1/4 0
(A, Aa) 0 1/4 0
(A, aa) 0 0 0
(a, AA) 0 1/4 0
(a, Aa) 0 1/4 1
(a, aa) 0 0 0
Example 2: X-linked Inheritance
Parents(A, AA) (A, Aa) (A, aa) (a, AA) (a, Aa) (a, aa)
Offspring
(A, AA) 1 1/4 0 0 0 0
(A, Aa) 0 1/4 0 1 1/4 0
(A, aa) 0 0 0 0 1/4 0
(a, AA) 0 1/4 0 0 0 0
(a, Aa) 0 1/4 1 0 1/4 0
(a, aa) 0 0 0 0 1/4 1
Example 2: X-linked Inheritance
Solve where
Example 2: X-linked Inheritance
Solve where
Example 2: X-linked Inheritance
𝟎𝒏→∞
Example 2: X-lined Inheritance
Example 2: X-linked Inheritance
𝑥(𝑛)→[1 2 /3 1 /30 0 00 0 0
2 /3 1/3 00 0 00 0 0
0 0 00 0 00 1/3 2 /3
0 0 00 0 01/3 2/3 1
]𝑥(0)
Example 2: X-linked Inheritance
Initial Parents , i.e. , then
