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

提案：论文题目：分歧理论和延迟方程：在控制模式形成和蛋白质翻译建模中的应用摘要本研究提出了分歧理论和延迟微分方程的两个应用：（1）振荡模式的自动调节反馈控制，（2）细胞蛋白质翻译的数学建模。控制模式的研究提出了一种自调整反馈控制方案，旨在稳定振荡模式状态。反馈控制方法以这样一种方式利用目标图案的对称性，即当实现控制时，它变得非侵入性。两个相关的案例研究将继续进行：（A）在Benjamin-Feir不稳定区域的二维复杂Ginzburg-Landau方程的行波解的稳定化，和（B）在振荡区域的光敏Belousov-Zhabotinsky反应的化学行波图案的控制。这两个案例研究之间的数学关系将阐明所提出的分析。 蛋白质翻译的数学建模的研究是旨在通过系统近似推导出蛋白质翻译的延迟方程模型，该模型可以作为涉及多个蛋白质的合成基因网络的简单模型的一个组成部分。延迟模型是从拉伸过程的连续描述中得到的，拉伸过程最终在简化的数学模型中表现为延迟时间。这项研究将扩展该模型以纳入mRNA的降解。在简单的基因开关和振荡器的情况下，延迟模型的保真度的机制之一，然后将使用分歧理论，aidedby一个数值延拓包，是为delaydifferential equations.拟议的研究将有助于研究生和博士后研究员在跨学科的应用数学研究的培训。这将有助于消除化学反应扩散系统以及其他模式形成系统中的时空混沌的反馈控制方案的发展。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，即核糖体利用信使RNA（mRNA）中编码的信息一次一个氨基酸地组装蛋白质的过程。拟议的研究将有助于发展一个systematicmathematical框架推导减少延迟模型从复杂的，生物详细的机制模型。所提出的项目代表了延迟微分方程及其分支理论分析的重要应用。对这些案例的分析对于这些数学和计算工具的开发是必不可少的。

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.