一类扩充Goodwin振子的动力学行为+源代码
毕业论文关键词：扩充Goodwin振子；合作结合；基因调控；Hopf分支；振荡
The Dynamics of a Kind of Extended Goodwin Oscillator
Abstract：The two genes are cross-regulated by means of activation or repression, in which the translation product of one of the two genes has a positive catalytic effect on the substrate of the enzyme catalysis, and the regulation between the two genes and the substrate forms the expanded Goodwin oscillator that we will discuss. In the case of Hill coefficient up to 3, the fixed point of the expanded Goodwin oscillator in the form of the root of the higher order polynomial is searched by means of mathematical analysis and computer simulation. According to the Jacobian matrix, the regulatory determinant D is derived and the sign and size of the determinant D determine the stability and dynamics of the system analysis, and also determine whether the system oscillation and where the oscillation occurs. When the two gene subsystems are activation and repression conditions, the oscillation analysis of this type of expanded Goodwin oscillator are: (i) system with n<3, found no oscillation phenomenon. (2) system with n=3, for combination of activation and repression , we found the system sustain a Hopf bifurcation and undamped oscillation. In the scenario setting of this article, the regulation determinant D only adopts the positive sign.
Key words：expanded Goodwin oscillator; cooperative binding; gene regulation; Hopf bifurcation; oscillation
目 录
摘要2
关键词2
Abstract…2
Key words2
引言…2
1 动力学方程…4
1.1 结合平衡 …4
1.2 反应动力学 …5
2 定性分析…6
2.1 不动点的确定…6
2.2 Jacobi矩阵…7
3 结果与分析 10
3.1 非合作结合…10
3.2 合作结合…13
讨论…18
致谢…19
参考文献19
一类扩充Goodwin振子的动力学行为
引言基因调控的理论工作可以追溯到20世纪60年代，Jacob[1]和Monod[2]发现了第一个阻抑蛋白。之后，Goodwin发表了关于基因调控的振荡状态的第一篇论文[3]。人们对基因调控及其数学分析的兴趣从未停止[4-6]，关于基因调控网络模型的设计，进行了各种各样的不同的尝试，这些模型可用于系统生物学的遗传和新陈代谢网络的计算机模拟。然而，大多数模型对系统动力学行为的描述都非常简洁，为了对细胞动力学提供更好的理解，我们对细胞中的大网络的基本控制函数作出了解释。对于基因调控网络的研究，目前使用的数学方法包括布尔逻辑的应用[7-9]、随机过程[10]和确定性动态模型[11-13]。生物体内的结构和生物活体内的实验[14-18]不仅对动态调控提供了更深层次的认识，还对遗传和新陈代谢网络提供了更好的理解。除了各种简洁模型[19]，相对来说，只有较少的研究考虑了基于正反馈环和负反馈环[7,20]的动力学行为，如稳定性、双稳性、周期性或振荡。