對于DNA序列，一階馬爾科夫鏈可以理解為當前堿基的類型僅取決于上一位堿基類型。如圖1所示，一條序列的開端（由B開始）可能是A、T、G、C四種堿基（且可能性相同，均為0.25），若序列的某一位是A，則下一位堿基是A、T、G、C的概率分別為0.25、0.20、0.20、0.20，下一位無堿基（即序列結束，狀態(tài)為E）的概率為0.15。

圖1 DNA序列的一階馬爾科夫鏈

2. 代碼實現(xiàn)

以下代碼運行于Jupyter Notebook (Python 3.7)；代碼功能是隨機生成一定數(shù)量的DNA序列，統(tǒng)計序列長度并繪制分布圖。若希望顯示隨機生成的序列，將代碼# print(''.join(Seq))前的#刪除即可。

import numpy
import random
import seaborn as sns
import matplotlib.pyplot as plt

# 狀態(tài)空間
states = ["A","G","C","T","E"]

# 可能的事件序列
transitionName = [["AA","AG","AC","AT","AE"],
                  ["GA","GG","GC","GT","GE"],
                  ["CA","CG","CC","CT","CE"],
                  ["TA","TG","TC","TT","TE"],]

# 概率矩陣（轉移矩陣）
transitionMatrix = [[0.25,0.20,0.20,0.20,0.15],
                    [0.20,0.25,0.20,0.20,0.15],
                    [0.20,0.20,0.25,0.20,0.15],
                    [0.20,0.20,0.20,0.25,0.15]]

def RandomDNAs(Num):
    max_len = 0
    i = 0
    Seq = [] #創(chuàng)建列表(Seq)用于添加堿基，以組成DNA序列
    Len = [] #創(chuàng)建列表(Len)用于記錄每條生成序列的長度
    while i != Num:
        Base = ["A","G","C","T"]
        START = random.choice(Base) #隨機從堿基中選擇一個作為序列的起始堿基
        Seq.append(START) #將起始堿基添加至Seq中
        while START != "E":
            if START == "A":
                change = numpy.random.choice(transitionName[0],p=transitionMatrix[0])
                #以transitionMatrix矩陣第一行的概率分布隨機抽取transitionName第一行包含的事件
                if change == "AA":
                    START = "A" #如果轉移狀態(tài)是AA(即A堿基接下來的堿基是A，則將起始堿基設為A)
                elif change == "AG":
                    START = "G"
                elif change == "AC":
                    START = "C"
                elif change == "AT":
                    START = "T"
                elif change == "AE":
                    START = "E"
            elif START == "G":
                change = numpy.random.choice(transitionName[1],p=transitionMatrix[1])
                if change == "GA":
                    START = "A"
                elif change == "GG":
                    START = "G"
                elif change == "GC":
                    START = "C"
                elif change == "GT":
                    START = "T"
                elif change == "GE":
                    START = "E"
            elif START == "C":
                change = numpy.random.choice(transitionName[2],p=transitionMatrix[2])
                if change == "CA":
                    START = "A"
                elif change == "CG":
                    START = "G"
                elif change == "CC":
                    START = "C"
                elif change == "CT":
                    START = "T"
                elif change == "CE":
                    START = "E"
            elif START == "T":
                change = numpy.random.choice(transitionName[3],p=transitionMatrix[3])
                if change == "TA":
                    START = "A"
                elif change == "TG":
                    START = "G"
                elif change == "TC":
                    START = "C"
                elif change == "TT":
                    START = "T"
                elif change == "TE":
                    START = "E"
            if START != "E":
                Seq.append(START) #如果狀態(tài)轉移后不為End(E)，則將轉移后的堿基加到Seq序列中
        i += 1
        Len.append(len(Seq))
        if len(Seq) &gt; max_len:
            max_len = len(Seq)
        #print(''.join(Seq))
        Seq.clear()
    plt.hist(numpy.array(Len), bins=max_len, edgecolor="white")
    # 顯示橫軸標簽
    plt.xlabel("DNA Sequence Length")
    # 顯示縱軸標簽
    plt.ylabel("Frequency")
    # 顯示圖標題
    plt.title("Histogram of frequency distribution of DNA sequence length")
    plt.show()
    print("DNA序列的最大長度為:",max_len)
    print("DNA序列長度的眾數(shù)為:",max(Len, key=Len.count))

%matplotlib notebook #若未使用Jupyter Notebook，此句不需要
RandomDNAs(1000) #1000表示隨機生成1000條序列