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:

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