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