Bifurcation theory and delay equations: applications to controlling pattern formation and modeling protein translation

项目申请：DMS - 0709232 PI: Silber， MaryInstitution: Northwestern university题目：分岔理论和延迟方程：在控制模式形成和蛋白质翻译建模中的应用摘要本研究主要针对分岔理论和延迟微分方程的两个应用：(1)振荡模式的自调整反馈控制，(2)细胞蛋白质翻译的数学建模。在控制模式的研究中，提出了一种旨在稳定振荡模式状态的自调整反馈控制方案。反馈控制方法利用目标模式的对称性，使其在实现控制时变得非侵入性。将进行两个相关的案例研究：(A)在benjamin - feir不稳定状态下二维复杂Ginzburg-Landau方程平面波解的稳定，以及(B)在振荡状态下光敏Belousov-Zhabotinsky反应的化学行波模式的控制。这两个案例之间的数学关系将通过提出的分析来阐明。提出的蛋白质翻译的数学建模研究旨在通过系统近似推导蛋白质翻译的延迟方程模型，该模型可以用作涉及多个蛋白质的合成基因网络的简单模型的组成部分。延迟模型是由延长过程的连续统描述得到的，该过程最终在简化的数学模型中表现为延迟时间。拟议的研究将扩展该模型以纳入mRNA的降解。在简单基因开关和振子的情况下，延迟模型对机械模型的保真度将使用分岔理论进行研究，并辅以为延迟微分方程开发的数值延拓包。拟议的研究将有助于培养跨学科应用数学研究的研究生和博士后。这将有助于反馈控制方案的发展，以消除化学反应-扩散系统中的时空混沌，以及其他模式形成系统。延迟微分方程经常出现在生物过程的建模中，例如细胞蛋白质翻译，核糖体利用信使RNA （mRNA）编码的信息一次一个氨基酸组装蛋白质的过程。提出的研究将有助于开发一个系统的数学框架，用于从复杂的、生物详细的机制模型中推导出减少的延迟模型。提出的项目代表了延迟微分方程及其分岔理论分析的重要应用。对这些提出的案例研究的分析对这些数学和计算工具的发展至关重要。

Proposal: DMS - 0709232 PI: Silber, MaryInstitution: Northwestern UniversityTitle: Bifurcation theory and delay equations: applications to controlling pattern formation and modeling protein translation ABSTRACTThe proposed research addresses two applications of bifurcation theoryand delay differential equations: (1) autoadjusting feedback controlof oscillatory patterns, and (2) mathematical modeling of cellularprotein translation. The proposed research on controlling patternsinvestigates an autoadjusting feedback control scheme aimed atstabilizing oscillatory patterned-states. The feedback control methodexploits symmetries of the targeted pattern in such a way that itbecomes noninvasive when control is achieved. Two related case studieswill be pursued: (A) stabilization of traveling plane wave solutionsof the two-dimensional complex Ginzburg-Landau equation in theBenjamin-Feir unstable regime, and (B) control of chemical travelingwave patterns of the photo-sensitive Belousov-Zhabotinsky reaction inthe oscillatory regime. The mathematical relationship between thesetwo case studies will be elucidated by the proposed analysis. Theproposed research on mathematical modeling of protein translation isaimed at deriving, by systematic approximation, a delay equation model ofprotein translation that could then be used as a component in simplemodels of synthetic gene networks involving more than one protein. Thedelay model is obtained from a continuum description of the elongationprocess, which ultimately shows up as a delay time in the reducedmathematical model. The proposed research will extend the model toincorporate the degradation of mRNA. The fidelity of the delay modelto the mechanistic one, in the case of simple gene switches andoscillators will then be investigated using bifurcation theory, aidedby a numerical continuation package that was developed for delaydifferential equations.The proposed research will contribute to the training of graduatestudents and postdoctoral fellows in interdisciplinary, appliedmathematics research. It will aid the development of feedback controlschemes for eliminating spatio-temporal chaos in chemicalreaction-diffusion systems, as well as other pattern-forming systems.Delay differential equations frequently arise in the modeling ofbiological processes, such as cellular protein translation, theprocess whereby ribosomes assemble proteins, one amino acid at a time,using the information encoded in the messenger RNA (mRNA). Theproposed research will contribute to the development of a systematicmathematical framework for deriving reduced delay models from complex,biologically-detailed mechanistic models. The proposed projectsrepresent important applications of delay differential equations andtheir analysis using bifurcation theory. The analysis of theseproposed case studies are essential to the development of thesemathematical and computational tools.