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

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

• 批准号: 0709232
• 负责人: Mary Silber
• 金额: $ 38.34万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0709232&HistoricalAwards=false
• 关键词: Bifurcation; theory; delay; equations; applications

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.

建议：DMS-0709232 PI：Silber，Mary西北大学研究所题目：分叉理论和时滞方程：在控制模式形成和模拟蛋白质翻译方面的应用所提出的研究涉及分叉理论和时滞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。控制模式的研究提出了一种以稳定振荡模式状态为目标的自调整反馈控制方案。反馈控制方法以这样一种方式描述目标图案的对称性，即当实现控制时，它变得非侵入性。将继续进行两个相关的案例研究：(A)二维复Ginzburg-Landau方程在Benjamin-Feir不稳定区的行波解的稳定性；(B)在振荡区对光敏Belousov-Zhabotinsky反应的化学行波模式的控制。拟议的分析将阐明这两个案例研究之间的数学关系。蛋白质翻译的数学模型研究旨在通过系统的近似推导出蛋白质翻译的延迟方程模型，然后将其作为涉及多个蛋白质的合成基因网络的简单模型的组成部分。延时模型是从拉伸过程的连续描述中得到的，在简化的数学模型中最终表现为延时。这项拟议的研究将扩展该模型，以包括mRNA的降解。在简单的基因开关和振子的情况下，延迟模型与机械模型的保真度将用分叉理论进行研究，并辅之以为延迟微分方程开发的数值延续程序包。所提出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。它将有助于开发反馈控制方案来消除化学反应-扩散系统以及其他模式形成系统中的时空混沌。在生物过程的建模中，经常会出现时滞微分方程，如细胞蛋白质翻译，即核糖体利用信使RNA(MRNA)中编码的信息一次组装蛋白质的过程。拟议的研究将有助于开发一个系统的数学框架，用于从复杂的、生物学上详细的机制模型中推导出减少延迟的模型。所提出的方案代表了时滞微分方程及其分支理论分析的重要应用。对这些拟议的案例研究的分析对于开发这些数学和计算工具是必不可少的。