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Stability and Bifurcations in Free-Boundary Models of Active Gels

活性凝胶自由边界模型的稳定性和分岔

基本信息

批准号：
2005262
负责人：
Leonid Berlyand
金额：
$ 28.5万
依托单位：
Pennsylvania State Univ University Park
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005262&HistoricalAwards=false
关键词：
Stability Bifurcations Free Boundary Models

项目摘要

This project is motivated by studies of active matter (also known as active materials). Such materials are typically of biological origin, such as bacterial suspensions and cytoskeletons of living cells, but they also include synthetic systems such as artificial self-propelled particles. These materials exhibit striking novel properties, and their theoretical understanding requires development of new mathematical tools. A signature of many active materials is motility, which is the ability to move spontaneously via the consumption of energy from internal sources or from the environment. This project concerns the development and analysis of mathematical models of a special class of active material: motile active gels in a non-equilibrium state (for example, the cytoskeleton of a living cell). The project focuses on free-boundary models, in which unknown functions solve equations in a domain that, due to the phenomenon of motility, is also unknown. The methodology under development will be applicable in applied mathematics and will be relevant to materials and life sciences as well as engineering. The graduate students participating in this project will receive highly multidisciplinary training, enabling them to work at the interface of mathematics, life, and physical sciences. The principal investigator will also teach and mentor undergraduate students with an emphasis on applications of mathematics to other disciplines. Free boundary problems such as the Stefan, Hele-Shaw, and Muscat problems have received significant attention since they are challenging from a mathematical point of view and important for applications. The focus of this project is on rigorous analysis of recently developed free-boundary partial differential equation (PDE) models of active gels (gels in a non-equilibrium state). These models are governed by nonlinear PDEs, unlike classical free boundary problems and most recent tumor growth models. The principal investigator will perform rigorous analysis of the existence and stability of stationary state, traveling wave, and rotating solutions by developing novel analytical tools for bifurcation analysis in the free boundary setting. The stability of these solutions is crucial for biophysical applications since it allows one to distinguish stable states from unstable ones that are rarely observed in experiments. The project will demonstrate that linearized stability analysis of these solutions is typically inconclusive, and new techniques for genuine nonlinear stability analysis will be developed based on construction of novel energy (Lyapunov type) functionals for free boundary problems with nonlinear PDEs. Analytical and numerical results will be compared to experimental observations of single cells and cell aggregates. A long-term goal is to provide theoretical understanding of migration of cells that drives important biological processes, for example, wound healing and the invasion of cancerous tissues.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目的动机是研究活性物质（也称为活性材料）。这些材料通常是生物来源的，如细菌悬浮液和活细胞的细胞骨架，但它们也包括合成系统，如人工自推进颗粒。这些材料表现出惊人的新特性，它们的理论理解需要开发新的数学工具。许多活性材料的特征是运动性，这是通过消耗来自内部来源或环境的能量而自发移动的能力。该项目涉及开发和分析一类特殊活性材料的数学模型：非平衡状态下的能动活性凝胶（例如活细胞的细胞骨架）。该项目的重点是自由边界模型，其中未知函数求解域中的方程，由于运动现象，也是未知的。正在开发的方法将适用于应用数学，并将与材料和生命科学以及工程有关。参加该项目的研究生将接受高度多学科的培训，使他们能够在数学，生命和物理科学的接口工作。首席研究员还将教授和指导本科生，重点是数学在其他学科的应用。自由边界问题，如Stefan，Hele-Shaw和麝香问题，受到了极大的关注，因为它们是具有挑战性的从数学的角度来看，重要的应用。该项目的重点是严格分析最近开发的自由边界偏微分方程（PDE）模型的活性凝胶（凝胶在非平衡状态）。这些模型是由非线性偏微分方程，不像经典的自由边界问题和最近的肿瘤生长模型。首席研究员将通过开发用于自由边界设置中的分叉分析的新型分析工具，对稳态、行波和旋转解的存在性和稳定性进行严格分析。这些解决方案的稳定性是至关重要的生物物理应用，因为它允许一个区分稳定的状态，从不稳定的，很少在实验中观察到的。该项目将证明，这些解决方案的线性化稳定性分析通常是不确定的，真正的非线性稳定性分析的新技术将开发基于新的能量（李雅普诺夫型）泛函与非线性偏微分方程的自由边界问题的建设。分析和数值计算的结果将进行比较，单细胞和细胞聚集体的实验观察。长期目标是提供对细胞迁移的理论理解，这些迁移驱动重要的生物过程，例如伤口愈合和癌组织的侵袭。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。