# Discussion on the exercise of computational molecular evolution – Chapter 1

Time：2022-1-23

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:     so Get 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 eigenvalue Simplify P (T), where   take Substitute in, get In jc69 model, the total replacement rate of any nucleotide I is , recorded as  Namely Solution , therefore ## A front-end developer's Vim is the same as an IDE

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