MAT 4830 Mathematical Modeling 4.4 Matrix Models of Base Substitutions II

Markov Models Review of Eigenvalues and Eigenvectors An example a Markov model. Specific Markov models for base substitution: Jukes-Cantor Model Kimura Models (Read)

Recall Characteristic polynomial of A Eigenvalues of A Eigenvectors of A

Recall Characteristic polynomial of A Eigenvalues of A Eigenvectors of A

Geometric Meaning

Lemma

Recall Use the transition matrix, we can estimate the base distribution vectors of descendent sequences by An example of Markov model

Markov Models Assumption What happens to the system over a given time step depends only on the state of the system and the transition matrix

Markov Models Assumption What happens to the system over a given time step depends only on the state of the system and the transition matrix In our case, p k only depends on p k-1 and M

Markov Models Assumption What happens to the system over a given time step depends only on the state of the system and the transition matrix In our case, p k only depends on p k-1 and M Mathematically, it implies

Markov Matrix All entries are non-negative. column sum = 1.

Markov Matrix : Theorems Read the two theorems on p.142

Jukes-Cantor Model

Additional Assumptions All bases occurs with equal prob. in S 0.

Jukes-Cantor Model Additional Assumptions Base substitutions from one to another are equally likely.

Jukes-Cantor Model

Observation #1

Mutation Rate Mutation rates are difficult to find. Mutation rate may not be constant. If constant, there is said to be a molecular clock More formally, a molecular clock hypothesis states that mutations occur at a constant rate throughout the evolutionary tree.

Observation #2

The proportion of the bases stay constant (equilibrium) What is the relation between p 0 and M?

Example 1 What proportion of the sites will have A in the ancestral sequence and a T in the descendent one time step later?

Example 2 What is the prob. that a base A in the ancestral seq. will have mutated to become a base T in the descendent seq. 100 time steps later?

Example 2 What is the prob. that a base A in the ancestral seq. will have mutated to become a base T in the descendent seq. 100 time steps later?

Example 2

Homework Problem 1

Example 2 (Book’s Solutions)

Our Solutions

Maple: Vectors

Homework Problem 2 Although the Jukes-Cantor model assumes, a Jukes-Cantor transition matrix could describe mutations even a different. Write a Maple program to investigate the behavior of.

Homework Problem 2

Homework Problem 3 Read and understand the Kimura 2- parameters model. Read the Maple Help to learn how to find eigenvalues and eigenvectors. Suppose M is the transition matrix corresponding to the Kimura 2-parameters model. Find a formula for M t by doing experiments with Maple. Explain carefully your methodology and give evidences.

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