Research On Stability Of Delayed Genetic Regulatory Network

Genetic regulatory network is one very important sort of complex networks, it lies at the heart of the signal processing functionality in real lives. A major challenge in the post-genomic era is to quantitatively understand topological and dynamical properties and functions of these complex biological networks that control cellular functions. As is well known, the introduction of time delay often leads to great effects on the dynamical behavior in the fields of network systems. But the time delay has to be considered into the factors which influence systems due to the limitation of the signal transformation velocity. The delayed genetic regulatory network is an important part of the delayed large-scale system, which exhibits the rich and colorful dynamical behaviors. At the same time, gene regulation is an inherently noisy process, thus, noise on the impact of gene regulatory networks are also factors that must be considered. This paper mainly discussed the stability of genetic regulatory networks from two impact factors, time delay and noise, and obtained some interesting results.The main contents and originalities in this dissertation can be summarized as follows:â‘ Robust stability of genetic regulatory network with time-varying delays and noise perturbation.Due to the fact that the slow process of transcription and translation will inevitablely lead to time delays, and the random births and deaths of individual molecules and environment fluctuations make gene regulation process full of noises. We study the robust stability of delayed genetic regulatory network. Based on Lyapunov stability theory and Lur's system method, novel delay-dependent and rate-dependent robust stability criteria of genetic regulatory network are derived.â‘¡Exponential stability of stochastic genetic regulatory network with time-varying delays.As the time-varying delay can be considered a function of time t, then the function may be differentiable or non-differentiable. Exponential stability of stochastic genetic regulatory network with differentiable time delay and non-differentiable time delay are studied respectively, and delay-dependent and rate-dependent,delay-dependent and rate-independent exponential stability criteria are separately derived in the form of linear matrix inequalities. Since it is meaningful for design of gene circuit that genetic regulatory network can rapidly converge to stable state under stochastic disturbances, therefore, the study on exponential stability of genetic regulatory network provides theoretical basis for design of gene circuit.â‘¢Asymptotical stability of stochastic genetic regulatory network with time-varying delays and distributed delays.Real genetic regulatory networks usually have a spatial extent because of plenty of parallel pathways with a variety of molecule size and length, which may result in distributed transmission delays. In this dissertation, asymptotical stability of stochastic genetic regulatory network with time-varying delays and continuous distributed delays is discussed. The two cases, differentiable time-varying delays and non-differentiable time-varying delays, are studied respectively, the limitation that the derivative of time-varying delays must be less than 1 is removed, the activation function is less restrictive, and noise is considered in transcription and translation process. Using Lyapunov-Krasovskii functional and stochastic differential method, we obtain sufficient criteria in the form of LMI. Literature dealing with stability for genetic regulatory network with mixed delays appears to be scarce. The analyzing method for the system's stability in this dissertation may be helpful to solve these problems.â‘£Stability analysis of coupled genetic regulatory network with time delays and parameter uncertainties.In uncertain genetic regulatory network, equilibrium point is unknown which is the solution of a system of parameter-dependent nonlinear equations. We study the effect of parameter uncertainties on stability of delayed genetic regulatory network. When parameter uncertainties occur, the delayed genetic regulatory network is stable in a region which contains the equilibrium point of nominal system. Using Lyapunov-type stability theory, new stability criteria of delayed genetic regulatory network are obtained, and the bound of converging region is also estimated. The results and analytical method can provide theoretical basis for the study on robust stability of genetic regulatory network with unknown equilibrium point.