| Molecular biology in the 20th century experienced the developmental processes from macroscopical to microscopical cosmic, i.e., from morphological and phenotypic description to the study of the interactions between various molecules in organisms, systems biology studies the interactions between molecules with various structures and functions from the levels of cell, tissue, organ and the whole organism, quantitatively describes and predicts biological function, phenotype and behavior by theoretical and computational biology, systems biology research is a gradually integrative process, so it is always be called the biology of 21th century.Under the dual effects of the individual interactions and the variations of external environment, organisms may exhibit many new global properties, such as multistability, bifurcation, adaptation and memory etc. According to the nonlinear dynamical modeling and analysis, we can obtain various new insights and understanding of the structures and functions of organisms. The complexity of organism and many nonlinear dynamical properties of biological processes is a new challenge to both applied mathematics and computational biology.40 years ago, Monod and Jacob made a bold prediction that basic cellular pro-cesses are accomplished through signal pathways at the level of genes. Such a prediction has laid a foundation for describing many detailed and gene-level important regulation mechanisms. Since natural regulatory networks are too complicated, they still lack math-ematical model-based quantitative descriptions at the present stage. With development of nonlinear theory and appearance of mainframe computer, however, theoreticians put forwards a lot of theoretical models describing biological networks successively. In par-ticular, with advances in biological experiments and bioengineering technology, it is possible to develop some circuit-based analysis technologies to describe complicated regulatory networks, allowing for the description of models underlying basic cellular functions during gene regulatory processes. According to achievements in fields of non-linear theory and stochastic process, developing quantitative and qualitative analysis of biological networks including gene regulatory networks is timing.There are lots of network motifs that are independent of each other but have partic-ular biological functions. Unraveling the functions of these simple motifs at the molec-ular level is of important significance for understanding the regulation mechanisms of more complicated regulatory networks and even for understanding intracellular pro-cesses, and is also helpful for elucidating design principles of biological networks. Our research method is based on the "Bottom-Up" method in systems biology and this the-sis studies the functions of several typical classes of network motifs from viewpoints of gene regulation, where feedback impenetrates all network motifs to be studied. For example, the positive and negative feedback loops appearing in Chapter 2 where their multistability and oscillation have been studied, one mutual double-negative feedback with auto-feedback regulatory loop in Chapter 3 where the switches between multiple steady states have been discussed, in Chapter 4 where the adaptation to periodic pulse stimulation of EFFL and NFL motifs motif have been studied. From viewpoints of sys-tems biology, we mainly study the dynamics properties of these functional motifs, e.g., multistability, oscillation, adaptation and response to stimulation etc.The main content of this thesis is organized as follows. In Chapter1, we first give a simple introduction about the related biological contexts and production background, research development, research contents and research methods about the newly rising in-tersectional subject-systems biology. Then we import the related basic knowledge about the gene expression regulatory networks, in the consequence we discuss dynamical mod-eling and analysis of the gene regulatory networks and the research aim and meaning elucidated in conclusion.In Chapter2, we study the effects of multiple parameters variation on biological system behaviors,and the dynamical behaviors of the corresponding gene regulatory net-works, such as multistabiliy and oscillations etc. From the viewpoints of control the-ory and systems biology, we use mathematical methods to analyze and conclude some prospective properties about the positive and negative feedback loops. In order better to obtain the organism designing principles, the importance of parameters in determining systems behavioral processes has to be estimated, analytical analysis about the effects of multiple parameter variations on biological network properties is given in chapter2. In this chapter, analyze network characteristics according to expand the monotone systems theory and lay out the effects of multiple parameter variations on system behaviors. This method is accomplished based on decomposing a closed loop system to several open monotone subsystems or modules. We clarify the methods provided are appropriate to general biological networks through a five-variable system and a oscillatory network.In chapter3, we use a simplified two-component network to analyze directed dif-ferentiation of stem cells, according to the system parameters perturbations representing the complex external and internal factors to clarify our analysis. These methods and conclusions will provide practical guidelines for study about the molecular mechanisms of cell differentiation, flexible control of the directed differentiation and developing new therapies for diseases treatment.In Chapter4, we discuss two kinds of simple networks which show adaptation to constant external inputs, and study their adaptation to time-dependent periodic exter-nal stimulus. Through analytical calculations and numerical simulations, we find they will show adaptation when proper pulse durations and internals adjusted. Especially for incoherent feedforward loops, two kinds of mathematical models-logic approximation piecewise linear equations and Hill function equations are considered. When consider the adaptation to time-dependent stimulus, the results from the analytical solutions of the logic approximation piecewise linear equations and numerical simulations of Hill function equations are completely consistent. Researches about the adaptation on time- dependent pulse stimulations will provide various possible solutions and instructions for developing effective therapeutic schedules and constructing biological circuits with spe-cialized functions. In the last chapter, we give a summary for the whole thesis, and prospect the direction on our scientific research.
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