臺灣博碩士論文加值系統

本篇論文的內容著重於研究基因調控網路中、轉錄和轉譯的過程內在雜訊的探討，不包含外在環境因素的影響。基因調控網路中蛋白質或訊息RNA的合成可以用rate equations來描述、此方程式為一巨觀之方程式。欲探討基因調控網路中內在雜訊的行為、首先我們的方法是從rate equations中的rate constants出發、再加上隨機過程的概念而寫出Master equation、進而利用Ω- expansion的技巧對平衡點做泰勒級數展開，將Master equation轉換成線性Fokker-Planck equation。
我們利用計算所得到的Fokker-Planck equation來分析與探討基因調控網路中蛋白質或訊息RNA濃度的起伏與變化。而蛋白質或訊息RNA濃度的變化與本篇論文中所探討的基因調控網路中內在雜訊有著密切的關聯性。基於此關聯性、我們再利用Fokker-Planck equation計算出variance和covariance。藉由此結果、我們將雜訊的強度用Fano Factor來表示、此定義為variance除以平衡點的值。
因此、藉由改變不同參數與所得到Fano Factor的值，我們去分析與探討不同參數對雜訊的影響。而此影響我們在論文裡有詳細的討論。

In this thesis, we investigate the intrinsic noise in transcription and translation level of genetic regulation networks without environmental conditions. The genetic regulation networks can be mathematically described by rate equations. To obtain the stochastic features of the system, the macroscopic rate equation is first rewritten as the stochastic master equation, and then expansion method is used to obtain the linear noise Fokker-Planck equation. We use the linear noise Fokker-Planck equation to analyze the stochastic fluctuations in the regulation networks of single gene and toggle switch. The noise strength is measured by Fano Factor which is defined as variance over mean, and the correlation of noise is also analyzed. The effect of regulation strength on the characteristics of noises and discussed.

Contents

Abstract (in Chinese) I
Abstract (in English) II
Acknowledgements III
Contents IV
List of Figures VI
List of Tables VIII

1 Introduction：Biological systems 1
1.1 Basic of DNA molecules 1
1.2 Transcription of genes to produce messenger RNA 3
1.3 Translation of protein 5
2 Genetic Regulation Networks 6
2.1 Control of gene expression 7
2.2 Basic gene expression 7
2.3 Regulation of single gene expression 9
2.4 Genetic toggle switch 10
2.5 Mathematical analysis of equilibrium stable points 11
3 Noise Propagation in Genetic Regulation Networks 14
3.1 The stochastic Master equation 15
3.2 Linear noise Fokker-Planck equation 16
3.3 Steady state statistics and noise propagation 19
3.4 Asymptotic solutions as t →∞ 24
3.5 Noise strength and correlation 25
4 Conclusion 26
4.1 Regulation network of single gene 26
4.2 Regulation network of toggle switch 27
Bibliography 36



List of Figures

Fig 1.1 The backbone of each DNA strand is contained of sugars and phosphates, and the four bases are A, T, G, C 2
Fig 1.2 Part of double strand DNA with A paired with T and G paired with C. Three
hydrogen bonds between G, C and two hydrogen bonds between A, T 3
Fig 1.3 (a) part of double strand DNA (b) pull strands apart in the transcription process (c) RNA strand is synthesized 4
Fig 1.4 Gene and its promoter 5
Fig 1.5 The central dogma of gene expression 5
Fig 2.1 An operon may have one or more genes which are transcribed into one messenger RNA. The operator region contains of repressor and activator binding region 7
Fig 2.2 Basic gene expression 8
Fig 2.3 Repressor binds on the operator site 9
Fig 2.4 Genetic toggle switch 10
Fig 2.5 Graphic showing three equilibrium points of genetic toggle switch. In equilibrium, we set up dx1/dt = dx2/dt=0 and find the equilibrium points. For α1 =156.25，α2 =15.6，β=2.5 and γ=1, we can obtain three equilibrium points. The three equilibrium points are point A (155.7634, 0.0995), point B (0.3324, 11.7080) and point C (1.3165, 6.7342). The point A and the point B are both stable with its eigenvalue are negative. The point C is an unstable point with its eigenvalue is positive 13
Fig 4.1 The Hill parameter β is increased and dissociation constant is fixed as Kd=800. The results of ν2 and R12 are both decreased 30
Fig 4.2 The dissociation constant Kd is increased and the Hill parameter is fixed as β=2. The results of ν2 and R12 are both increased 31
Fig 4.3 ν1 as a function of β and ν2 as a function of β, for the stable point A (155.76, 0.0995) 32
Fig 4.4 R12 as a function of β, for the stable point A (155.76, 0.0995) 33
Fig 4.5 ν2 as a function of β and ν2 as a function of β, for the stable point B (0.3324, 11.708) 34
Fig 4.6 R12 as a function of β, for the stable point B (0.3324, 11.708) 35



List of Tables

Table 1 Parameter values used for single-gene auto regulatory network 28
Table 2 Parameter values used for toggle switch 29

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