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研究生:許泰欣
研究生(外文):Tai-Hsin Hsu
論文名稱:病毒的DNA二維步行統計分析
論文名稱(外文):Statistical analysis of two-dimensional viral DNA walks
指導教授:楊緒濃
指導教授(外文):Su-Long Nyeo
學位類別:博士
校院名稱:國立成功大學
系所名稱:物理學系碩博士班
學門:自然科學學門
學類:物理學類
論文種類:學術論文
論文出版年:2003
畢業學年度:91
語文別:中文
論文頁數:102
中文關鍵詞:擴散係數DNA步行偏移量比值
外文關鍵詞:DNA walksDeviation ratiosDiffusion coefficients
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在這篇論文當中,我們介紹兩個統計方法分析21個動植物病毒的DNA二維步行。根據DNA序列中的鹼基的映射規則,這些二維步行被分為三個映射類別。為了能夠作對照,我們也考慮隨機數與無理數序列的二維步行。為了能夠描述DNA序列及隨機數與無理數序列的特性,而定義出兩個統計量。第一個我們所考慮的,是一個序列的擴散係數,是根據一個質點的隨機步行而定義出來的。第二個我們所考慮的統計量,是一個統計偏移量。在論文中,我們也概略的提到幾種常用的分析方法,包括功率譜分析、相關性分析、互訊息分析和赫斯特指數。功率譜中,病毒DNA所呈現的週期3特質,已經被解釋為和密碼子結構有關,而相關性函數及互訊息函數也是能看到DNA的週期性。赫斯特指數對於同種不同型的病毒DNA,也能提供一個分辨的方法。這些方法,都是提供一些特別的比較量,用來分辨不同物種間的差異,及生命體DNA在數學上所應具有的特質。
DNA 序列分析的幾種方法,通常是以一維步行做基礎。在這篇論文中,主要的工作是根據映射規則,以21個病毒DNA二維步行後所得的軌跡圖形,來計算其擴散係數及偏移量比值。這裡所提的擴散係數,並非DNA分子的擴散係數,而是借用統計上對於二維隨機步行的擴散係數定義,計算DNA二維步行軌跡所呈現的擴散係數。在擴散係數圖形中,我們取其最大值做比較,因此在正向序列或反向序列都可得到三種映射類別的擴散係數最大值。由於在某種映射類別所得到的二維步行軌跡,當它越是集中在同一方向,並且其圖形越接近一條直線時,其所得到的擴散係數最大值會越大。在我們所使用的二維映射中,正負X與Y軸皆是代表不同的鹼基。因此二維步行圖形是存在於某兩個鹼基方向軸中,這兩個鹼基方向軸所對應的鹼基,就是在DNA序列中占有比較大的比例。因此當某個映射類別,擴散係數最大值比較大時,表示有兩個鹼基所占的比例較大。對於一些病毒DNA序列在二維步行的圖形,並不容易看出其規則性,也就是和一般的隨機數或無理數序列的二維步行軌跡,很難分辨出它們之間的差異時,擴散係數的圖形卻可以比較出其差異。擴散係數最大值的比較,在21個動植物病毒DNA分析結果顯示,DNA長度必須大於5000,才能和隨機數或無理數序列作分辨。
在論文中,我們提出另一個計算量,那就是最大偏移量的比值。這個偏移量的數學定義,是參考相關性函數的定義。根據偏移量的圖形,取其絕對值的最大值和相對應的間隔參數得其比值,就是最大偏移量比值。它比擴散係數方法好的地方,在於生命體DNA序列和隨機數或無理數序列的比較,有更好的鑑別能力。在21個動植物病毒DNA分析結果顯示,當DNA長度在2000以上,就可以分辨出是否為生命體的DNA。這個方法可以得到和擴散係數最大值鹼基比例推論相同的方式,可以根據某兩個映射類別最大偏移量比值,得知可能是那兩種鹼基的比例較大。
以上所提的擴散係數最大值和最大偏移量比值,在這21個動植物病毒的分析結果當中,三個映射類別所得到的值,最大值大多落在類別(2)與(3)。依據擴散係數最大值或最大偏移量比值,在類別(2)與(3)中所得到的結果對於鹼基(C+G)%作圖,可以得到類似的關係圖結果。當(C+G)%小於50%,擴散係數最大值或最大偏移量比值,會隨著(C+G)百分比的增加而有指數性的遞減。當(C+G)%大於50%,擴散係數最大值或最大偏移量比值,會隨著(C+G)百分比的增加而有指數性的遞增。這個結果對於CG含量的相關研究,將能提供一些參考。
從病毒DNA序列,我們發覺出某種規則性的排列,特別是二維步行圖形是接近一條直線的行為。這現象代表某兩種鹼基占的比例較大。雖然擴散係數與偏移量比值,是從二維步行圖形中得到的,但對於鹼基如何在DNA序列中重複排列的訊息,仍是未知。
In this thesis, we introduce two statistical analysis methods for the two-dimensional walks for 21 DNA sequences of animal and plant viruses. The two-dimensional walks are defined by three groups of mapping rules for the nucleotides in the DNA sequences. For the purpose of comparison, two-dimensional walks for sequences of random numbers and irrational numbers are also considered. To describe the properties of the DNA sequences and the random-number and irrational-number sequences, two statistical quantities are defined. The diffusion coefficient for a sequence is defined according to the random walk of a point particle. The second statistical quantity we considered is a statistical deviation.
In this thesis, we first outline some common analysis methods. They include the power spectrum, correlation, mutual information and Hurst exponent. The power spectra of the viral DNA sequences are seen to have the 3-bp periodicity, which is related to the codon structure of DNA sequences. The correlation and mutual information can also give the periodicity of DNA sequences. The Hurst exponent can be used to distinguish different types of viral DNA sequences of the same species. All these methods provide particular quantities, which allow us to distinguish the differences between organisms and show the characteristics of the DNA sequences of the organisms.
Previous analysis methods for DNA sequences have been based on one-dimensional mappings. According to the mapping rules in this thesis, the major effort is to calculate the diffusion coefficients and the deviation ratios for the two-dimensional DNA walker diagrams of 21 viral DNA sequences. The diffusion coefficients here are not the real diffusion coefficients of the DNA molecules. We use the statistical definition of two-dimensional random walks for the viral DNA sequences based on two-dimensional mapping rules. We use the maximum diffusion coefficients with three mapping groups to compare the organisms. The normal or reversed sequences are considered for their maximum diffusion coefficients with three mapping groups. The maximum diffusion coefficients are large with certain mapping groups when the two-dimensional walker diagrams enjoy nearly straight-line paths in certain directions. We note that in the two-dimensional mappings, the types of bases are chosen to represent the positive or negative X andY axes. So the two-dimensional walker diagram lies between the two axes. It implies that the fractions of two of the four types of bases are comparatively larger in the DNA sequence. When a certain maximum diffusion coefficient is large, it shows which two possible types of bases have large fractions in the DNA sequence. For some two-dimensional viral DNA walker diagrams, it is not easy to distinguish them from those of random-number or irrational-number sequences with some mapping groups, but their diffusion coefficients are quite different. In particular, we can use the maximum diffusion coefficient to distinguish organisms'' DNA sequences from random-number sequence or irrational-number sequences. In the analysis results of the 21 plant and animal viral DNA sequences, if the total length of an organism''s DNA is greater than 5000 base pairs, then the difference between organism''s DNA sequence and random-number sequence or irrational-number sequences can clearly be seen.
Further, we consider another statistical quantity. It is the maximum deviation ratio. The deviation ratio is defined with reference to the definition of correlation. From the deviation diagrams, we define the maximum-deviation ratio, which is given by the ratio of the maximum absolute deviation value to the distance at which the maximum value occurs. This quantity is better than the maximum diffusion coefficient. The reason is that it provides better resolution to distinguish organisms'' DNA sequences from random-number sequence or irrational-number sequences. In the analysis results of 21 plant and animal viral DNA sequences, we see that if the total lengths of organisms'' DNA are greater than 2000 base pairs, then the difference between organisms'' DNA sequences and random-number sequence or irrational-number sequences can be seen. The maximum deviation ratios with different mapping groups also lead to the same results of the maximum diffusion coefficients that we can deduce which two possible types of bases have large fractions in the DNA sequence.
Since large maximum diffusion coefficients and large maximum deviation ratios generally occur with groups (2) and (3), we plot the maximum diffusion coefficients or maximum deviation ratios with mapping groups (2) and (3) versus (C+G)%. We can see that when (C+G)% is smaller than 50%, the maximum diffusion coefficients or maximum deviation ratios decrease exponentially with (C+G)%. When (C+G)% is larger than 50%, the maximum diffusion coefficients or maximum deviation ratios increase exponentially with (C+G)%. This result provides some reference to researches related to the CG content.
We can also find some regularly repeating group of bases in viral DNA sequences especially for two-dimensional walks of straight-line shapes. This feature indicates that there are two types of bases that have large fractions in the DNA sequence. Although the diffusion coefficients and deviation ratios can be obtained from the two-dimensional walks, the information about the repeating feature of bases in DNA sequences is not known.
第一章序論………………………………………………………… 1
1.1DNA的結構………………………………………………… 1
1.2GenBank的資料庫格式…………………………………… 6
第二章映射模型…………………………………………………… 16
2.1 一維映射…………………………………………………………… 16
2.2 二維映射…………………………………………………………… 17
2.3 三維映射…………………………………………………………… 18
第三章常用的分析方法…………………………………………… 19
3.1 功率譜……………………………………………………………… 19
3.2 相關性……………………………………………………………… 22
3.3 互訊息……………………………………………………………… 24
3.4 赫斯特指數………………………………………………………… 28
3.4.1 R/S分析方法……………………………………………………… 30
3.4.2 DFA分析方法…………………………………………………… 31
3.4.3 小波分析方法…………………………………………………… 39
第四章擴散係數…………………………………………………… 41
4.1 二維步行…………………………………………………………… 41
4.2 二維的擴散係數定義……………………………………………… 46
4.3 二維的擴散係數最大值的分析結果……………………………… 47
第五章最大偏移量比值…………………………………………… 66
5.1 二維步行…………………………………………………………… 66
5.2 最大偏移量比值的定義…………………………………………… 67
5.3 最大偏移量比值的分析結果……………………………………… 69
第六章結論………………………………………………………… 88
附錄A 冪定律指數關係的證明……………………………………… 89
附錄B X軸或Y軸直線步行的最大偏移量比值……………………… 92
附錄C y=x方向步行的最大偏移量比值……………………………… 93
參考文獻……………………………………………………………… 98
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