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

提案：DMS - 0709232 PI：Silber，Mary 机构：西北大学 标题：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 该研究解决了分岔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，以及（2）细胞蛋白质翻译的数学建模。所提出的控制模式研究研究了一种旨在稳定振荡模式状态的自动调节反馈控制方案。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无创。将进行两个相关的案例研究：（A）本杰明-菲尔不稳定状态下二维复数Ginzburg-Landau方程的行波平面波解的稳定性，以及（B）振荡状态下光敏Belousov-Zhabotinsky反应的化学行波模式的控制。这两个案例研究之间的数学关系将通过所提出的分析来阐明。 拟议的蛋白质翻译数学模型研究旨在通过系统近似推导蛋白质翻译的延迟方程模型，然后将其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。延迟模型是从伸长过程的连续描述中获得的，最终在简化的数学模型中显示为延迟时间。拟议的研究将扩展模型以纳入 mRNA 的降解。在简单基因开关和振荡器的情况下，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连续包。拟议的研究将有助于跨学科应用数学研究中研究生和博士后的培训。它将有助于反馈控制方案的开发，以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。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，即核糖体使用信使 RNA (mRNA) 中编码的信息一次组装一个氨基酸的过程。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、生物学详细的机械模型中导出减少延迟的模型。所提出的项目代表了时滞微分方程的重要应用及其使用分岔理论的分析。对这些拟议案例研究的分析对于这些数学和计算工具的开发至关重要。

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.