基因表达过程是生命的基本过程，DNA分子是绝大多数生物的遗传信息载体。即使在单细胞生物中，基因的表达和蛋白质的调控也是通过相互作用形成一个复杂的调控网络，实验表明通常呈现为非线性动力学行为。而由于细胞尺寸的有限，mRNA数目的有限以及周围环境如光照、温度、压力和浓度分布等的不均匀性和不确定性，导致生化反应发生的随机性，因此基因表达必须要考虑噪声的影响，噪声一般分为外源噪声和内源噪声。研究噪声对基因调控网络的影响，仍然是当代生命科学的前沿课题之一。正如我们所知，在基因表达过程中，能够呈现出双稳态的最简单系统就是正自反馈单基因调控网络，即只有一个基因，经表达和翻译后，生成的蛋白质二聚体反过来又促进了该基因的表达。这就是本文主要研究的系统。本文从数值模拟和理论分析两方面研究噪声对于双稳态系统的动力学机制的影响。 在第一章中，我们引入基因表达领域的最新研究，并介绍研究相关领域的研究现状，从而引出我们要研究的问题以及说明我们的问题研究的必然性和重要性。由于基因表达的随机动力学研究是一门比较新兴的学科，其方法和研究都必须符合实验结论，所以，我们的结果必须建立在之前实验和结论之上，从而才能更有价值。 在第二章中，我们简单介绍基因表达动力学的相关模型的建立和以及常用方法，从而建立起动力学方程，并为下面一章作铺垫。通过模型，我们发现系统是双稳态系统，即对于某些参数值，系统有两个稳定点和一个不稳定点，就对应着细胞的两种表达状态，高表达状态和低表达状态，而如果这时由于存在随机噪声，就能够导致两个状态之间的随机转移。因此，我们就需要研究噪声对系统的影响。 在第三章中，我们对第二章中的动力学方程加入随机外源噪声，分别考虑加入加性外源噪声和乘性外源噪声对系统的影响，并且介绍随机模拟方法和得到的结论。近几年来，该领域发展了越来越多不同的数学模型以及数值模拟方法，用于描述基因调控网络的调控机制和探究潜在的调控原理。我们在前一章的基础上，通过对动力学模型加入随机的加性噪声和乘性噪声，来研究其对整个系统双稳态的影响，即主要目标是说明在随机加性噪声和乘性噪声下，概率转移速率和首达时间怎样被最大转移速率影响。在有双稳定性的正自反馈单基因网络中，当存在加性噪声和乘性噪声时，我们通过Fokker-Planck 方程确定相应的双稳态的势函数，并研究概率转移速率和首达时间的变化情况。同时，本论文采用Denisov（2003年）的方法，计算机模拟和随机过程的知识探讨噪声对该基因网络系统的影响。通过在数学方法和模拟方法得到系统的统计学性质，我们发现两者是吻合的。主要结果有以下几方面：（1）随着最大转移速率的增加，有利于系统保持高表达状态。这说明增大转移速率，系统将像高表达状态过渡。因此，如果想要得到高表达状态的话，我们只需增加最大转移速率即可达到目标。（2）如果加性噪声强度增大的话，从一个势井到另一个势井的概率转移速率也会随之增加。这说明加性噪声和势井之间跃迁的相关性，如果增大加性噪声强度，势井之间的转移概率会增加，因此，如果想要快的转移速率的话，就需要加大加性噪声的强度。（3）随着乘性噪声强度的增加，到达左势井的概率会随之增加。这说明乘性噪声的强度有利于促进左势井的概率的增加。因此，如果我们想要得到低的表达状态的话，需要加大乘性噪声的强度即可。（4）如果加性噪声和乘性噪声是正相关的话，噪声将会增加到达左势井的概率；反之如果加性噪声和乘性噪声是负相关的话，噪声将会增加到达右势井的概率。因为噪声之间的相关性没有确定的结论，所以我们得到两种情况，如果正相关的话，会增大左势井概率；如果负相关的话，会增大右势井的概率。 由于双稳定性在生物系统中是广泛存在的，从噬菌体到哺乳动物，通过我们的研究，说明噪声是怎样促进是势井间的转移，为以后研究外源噪声的相关实验和进一步相关研究提供了理论依据。

The process of gene expression is the basic process of life. DNA is usually the carrier of genetic information for the vast majority organisms. Even in a single cell, gene expression and the regulation of proteins interact to form a complex regulatory network. Experiments show that it usually presents as a non-linear dynamics. Because of the limited size of cell, the limited number of mRNA and the surrounding environment such as light, temperature, pressure and concentration distribution of the inhomogeneity and uncertainty, it is inevitable that biochemical reactions are stochastic processes.Therefore, noise must be considered in gene expression, which is generally divided into external noise and internal noise. Research on the noise impact of gene regulatory networks is still one of the forefront topics for the contemporary life sciences. As we know, in the process of gene expression which can show a bistable system in the most simple feedback loop is the autoactivating positive-feedback loop. That is, only one gene expresses and translates into protein, and the generation of protein dimers, in turn, promote the expression of the gene. This is the system which we will discuss. Here, both of the numerical simulation and theoretical analysis are considerd in the stochastic dynamics for the bistable mechanism of the system. The thesis is arrangement as in chapter 1 we introduce some basic knowledge and background of our research in tend to present the recent research on the field of gene expression and lead to our research. Because the study of gene expression is the forefront discipline, so it is necessary for the current understanding of theoretical and experimental results. In chapter 2 we introduce our model and some basic background knowledge. By establishing a dynamic equation, we pave the way for the following chapter. Through the model, we found that the system is bistable systems, i.e., for some parameter values, the dynamic system has two stable states and one unstable state, which corresponds to the two expression states, high expression and low expression of the protein state. Due to random noise, it can lead to a random shift between two states. Experiments show that even for small noise, it may be enlarged in the process of gene expression regulation and may have a very significant impact on gene expression system. Thus, it is possible to change the whole system stability. Therefore, we need to study the impact of noise on the system. In chapter 3 we study the stability of our autoactivating positive-feedback loop in the presence of additive and multiplicative noises, including the potential, probability transition rate and the first-passage time by using Fokker-Planck equation. The adding random noise is introduced into our model and we consider the impact of the additive and multiplicative noises on the system. Our main object is to describe how the random additive and multiplicative noises effect the system. Stochastic simulation methods are used to analyze the system dynamics. In recent years, with the development of the field, more and more different mathematical models and numerical simulation methods are used to describe the regulatory mechanism of gene regulatory networks and to explore regulatory principles. In this thesis, we use the method by Denisov(2003), to simulate the stochastic process in the system of gene networks. In our research, the probability transition rate and first-passage time in an autoactivating positive-feedback loop with bistability are investigated, where the gene expression is assumed to be disturbed by both additive and multiplicative external noises. The bimodality in the stochastic gene expression is due to the bistability. And the bistability determines that the potential of the corresponding Fokker-Planck equation has two potential wells. We also study the steady-state statistical properties like expectation and variance of our model in mathematics and simulation. Our main goal is to illustrate how the probability transition rate and first-passage time are affected by the maximum transcriptional rate, the intensities of additive and multiplicative noises, and the correlation of additive and multiplicative noises. Our main results show that (i) the increase of the maximum transcription rate will be useful for maintaining a high gene expression level; (ii) the probability transition rate from one potential well to the other one will increase with the increase of the intensity of additive noise; (iii) the increase of multiplicative noise strength will increase the amount of probability in the left potential well; and (iv) positive (or negative) cross-correlation between additive and multiplicative noises will increase the amount of probability in the left (or right) potential well. Because the bistability in biological systems is widespread, from bacteriophage to mammals, our research shows how the noise is a potential for the transfer between wells in the present of noise, and provides experimental and theoretical basis for further experiment and study.