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研究生： 魏建豪
Jian-Hao Wei
論文名稱： 基因序列的k 字齊普夫子集解析
k-tuple Zipf m-Set analysis on DNA
指導教授： 李弘謙
Hoong-Chien Lee
口試委員:
學位類別： 博士
Doctor
系所名稱： 理學院 - 物理學系
Department of Physics
畢業學年度： 99
語文別： 中文
論文頁數： 123
中文關鍵詞： 高頻字 、排名 、字的發生頻率 、全基因序列 、語言 、齊普夫定律 、編碼區 、非編碼區 、外顯子 、內含子 、頻率分佈 、冪次分佈
外文關鍵詞： coding parts, high-frequency words, ranking, k-mers, frequency of occurrence of words, complete genome sequences, noncoding parts, Zipf’s law; natural language, exons, introns, power-law distribution, frequency distribution
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    • 一個普遍被使用的數理統計方法－齊普夫定律，1994年被Mantegna與他的研究團隊使用在基因序列k字串的發生頻率與其排名的解析上（k字串齊普夫解析)，強調非編碼區有類語言的冪次規則。不過，這樣的結論被大量的質疑與討論。
    我們整理不同的齊普夫分佈研究領域，發現觀察的重點雖不盡相同，但事件總數為N時，各別事件在隨機狀態時機率均為1/N。然而，基因序列在序列的p(序列A+T含量所佔比)越遠離一半時，各別字串的機率在隨機狀態差異越大，因此在非隨機狀態中，機率不等是受到p與生物特徵兩個因素造成，影響齊普夫分佈的解析判斷。
    這個研究中，我們運用不同p的基因體序列與其對應的隨機序列的數據，證實k字串齊普夫子集解析法可以去除p的影響，改善k字串齊普夫解析難以定義隨機序列冪次的障礙，確立子集解析的優勢。
    另外，我們擬合四個函式（直線、指數、對數、冪次）選定足以代表物種特徵的「高頻字」（高頻率出現的字串），並嘗試找出865個物種高頻字冪次的普適性。研究結果顯示物種的冪次與其物種複雜度有關，傳達基因複製的演化結果。

    Zipf’s law is a characterization of the relation between the frequency of any word in a text and the ranking of that word in the frequency table. It states that if the text is that of a natural language, then the frequency versus ranking relation is an approximate power law. For a few years in the mid to late 1990’s Zipf’s law was intensely discussed in the context of genomic sequences, but no clear consensus was reached as to whether, as a general rule, the word frequencies -- a genomic a word is an oligonucleotide of a given length; we call a k-nucleotide word a k-mer -- in genomic sequences, or some specific portion thereof, obey a Zipf’s law. Here we revisit the issue by studying the frequency versus ranking relations of a large number of complete genomes, and of parts of genomes having different biological functions. We show that the nucleotide composition has an influence on the frequency versus rank relation of a genomic sequence that is strong enough to mask whatever Zipf’s-law behavior the sequence may possess. Once this influence is removed, then all genomes obey the same broadly defined classes of Zipf’s laws, with the most important class-defining factor being the length of k-mers, or the integer k. For eukaryotes, the Zipf’s laws for the exonic and intronic segments of the genome differ significantly. Based on the observation that the Zipf’s law of a sequence is determined by the subset of k-mers having the highest frequencies (of occurrence), we derive a relation between the Zipf’s-law exponent and the high-frequency tail of the frequency distribution, and infer that for genomes in general the high-frequency tail is best represented by an exponential function, as opposed to linear, logarithmic, or power-law functions.

    摘要 .......................................................... i ABSTRACT ..................................................... ii 序 .......................................................... iii 誌謝 ......................................................... iv 1. 緒論(INTRODUCTION) ......................................... 1 1.1 生物訊息的載體 ...........................................................................................1 1.1.1 生命的起源..................................................................................................... 1 1.1.2 基因序列的構造............................................................................................. 2 1.2 基因序列的演化模式...................................................................................3 1.2.1 基因序列的突變與重組................................................................................. 3 1.2.2 自然選擇與物種分類..................................................................................... 5 1.3 隨機系統的特性...........................................................................................6 1.3.1 隨機的定義..................................................................................................... 6 1.3.2 中央極限定理................................................................................................. 7 1.4 齊普夫定律(Zipf law)與現象觀察..............................................................7 1.4.1 文字資訊的書目計量學(Bibliometrics)........................................................... 7 1.4.2 何謂齊普夫定律？ .......................................................................................... 7 1.4.3 基因體序列的N 字串齊普夫定律.................................................................. 8 1.4.4 蛋白質表現的似齊普夫規則.......................................................................... 9 1.4.5 齊普夫定律無所不在.................................................................................... 10 1.5 齊普夫分佈的特性與應用.......................................................................10 1.5.1 最小努力原則(Principle of Least Effort)造成齊普夫分佈的魯棒性(robust) 11 1.5.1.1 Furusawa 建立簡單濃度擴散模式，2003 年.........................................11 1.5.1.2 Ogasawara 遺傳漂變和自然選擇的演化理論模型，2009 年................ 12 1.5.1.3 Bernat 運用算法信息論，模擬城市人口變動，2010 年...................... 13 1.5.1.4 其他例子................................................................................................... 14 1.5.2 尺度不變性與其冪次ζ ................................................................................. 14 1.5.2.1 氙Xe 的熱核碎裂，碎片分佈的冪次成氣液相變新依據..................... 15 1.5.2.2 基因表現量最大似然數分佈的冪次觀察癌症分類............................... 15 1.5.2.3 都市人口分佈、森林資源規模分佈與優化........................................... 18 1.5.3 訊息定量的Shannon 熵H 與冗數R....................................................... 19 1.5.3.1 基因體序列非編碼的含量影響影響熵與冗數....................................... 19 1.5.3.2 基因序列的G+C 含量影響結果？ ......................................................... 20 1.5.4 序列模型中，齊普夫指數ζ與長程關聯指數α .......................................... 20 1.5.4.1 對照序列的長程關聯指數與齊普夫指數的邊界................................... 21 1.5.4.2 齊普夫與長短程關聯並沒有對等的關係............................................... 22 2. 材料與方法 (MATERIALS AND METHODS)......................... 24 2.1 完整的基因體序列 .....................................................................................24 2.2 基因序列的k 字串齊普夫子集解析法(k-tuple Zipf m-Set analysis).........24 2.2.1 滑動窗口與k 字串齊普夫解析法............................................................... 24 2.2.2 相對頻率....................................................................................................... 25 2.2.3 相對子集頻率.................................................................................................. 26 2.3 排名機率分佈直方圖 (Rank-Probability density function Histogram, RPDF Histogram) ..................................................................................................26 2.4 以2%為分界的高頻字與低頻字............................................................26 2.4.1 DNA 序列字串齊普夫子集圖與高頻字測試....................................28 2.4.1.1 齊普夫子集圖的函式測試....................................................................... 29 2.4.1.2 排名機率分佈(RPDF)的限制，以機率分佈(PDF)取代之................... 30 2.4.1.3 機率分佈的函式測試............................................................................... 31 3. 研究結果(RESULTS) ......................................... 35 3.1 不同 p 的基因體與對應隨機序列的3 字串齊普夫解析.........................35 3.1.1 齊普夫圖與齊普夫子集圖........................................................................... 35 3.1.2 隨機序列的齊普夫(子集)冪次................................................................... 37 3.1.3 排名機率分佈直方圖................................................................................... 37 3.1.4 隨機序列突顯齊普夫子集解析優勢........................................................... 37 3.2 以數學基礎比較相對頻率與相對子集頻率...........................................39 3.2.1 為何相對頻率的隨機序列有階梯狀？....................................................... 39 3.2.2 相對頻率的隨機序列k 字串有k+1 階梯................................................... 40 3.2.3 相對子集頻率的隨機序列只有一個階梯................................................... 41 3.3 齊普夫子集解析冪次的普適性...................................................................41 3.3.1 字串長度、物種分類與冪次關係............................................................... 43 3.3.2 序列長度、p 對解析冪次的影響................................................................ 43 3.3.3 依p 與長度範圍分成五個分類................................................................... 44 3.3.4 基因體序列、基因區、基因間隔區、外碼子、內碼子的齊普夫冪次........ 45 4. 討論(DISCUSSION) .......................................... 48 4.1 物種的冪次與演化關係...............................................................................48 4.2 相對子集頻率不受到序列的p 大小影響.................................................48 4.3 齊普夫子集圖的曲線.................................................................................48 4.3.1 低頻字的隨機性.....................................................................................48 4.3.2 對形式的分類無特別益處........................................................................... 49 4.4 齊普夫冪次與序列種類無關，與序列的p、長度有關..........................49 4.4.1 冪次無異於序列類，以長度log(L)=5.4, 6.2 當新分界編為九個分類.... 49 4.4.2 冪次在短序列中對p 有顯著的差異、對長度無特定大小依靠............... 50 4.4.3 物種的齊普夫冪次於不同類型序列探索................................................... 51 4.4.4 齊普夫冪次與序列種類無關....................................................................... 52 參考資料..................................................... 54 附表 ......................................................... 57

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