Recently, I was studying Mr. Yang Ziheng’s computational molecular evolution. For the exercises at the back of each chapter, I may not find the answers on the Internet. Therefore, I put my calculations here. If there are students to discuss together, I will deepen my progress.
Because of my limited level of mathematics, many problems may not be solved. Here are only the solved exercises.
Chapter 1: nucleotide replacement model
1. Use the conversion probability under jc69 model (formula (1.3)) to verify Chapman Kolmogorov theorem (formula (1.4)). Consider two cases: (1) I = t, j = t; （2）i=T，j=C。 For example, in (1), determine。
Solution: formula (1.3):, where
，
。
The first case:，
。
The second case:，
。
soGet a certificate.
2. Deduce the transformation probability matrix under jc69 model。 Using the results in section 1.2.3, set the eigenvalues obtained based on tn93 model in the rate matrix (formula (1.15))
, and feature vector based on jc69 model Q
。 Another scheme is to directly derive eigenvalues and eigenvectors from formula (1.1), and then use formula (1.17).
Solution: from the results of section 1.2.3,
Among them,，
，
。
According to eigenvalueSimplify P (T), where
，
，
takeSubstitute in, get
In jc69 model, the total replacement rate of any nucleotide I is, recorded as
，
，
Namely，
Solution, therefore
。