Stability and Bifurcations in Free-Boundary Models of Active Gels

这个项目是由对活性物质(也称为活性物质)的研究推动的。这类材料通常是生物起源的，如细菌悬浮液和活细胞的细胞骨架，但它们也包括人造自推进颗粒等合成系统。这些材料表现出惊人的新奇性质，它们的理论理解需要开发新的数学工具。许多活性物质的一个特征是能动性，它是通过消耗来自内部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的活性材料的数学模型：处于非平衡状态(例如，活细胞的细胞骨架)的可移动的活性凝胶。该项目专注于自由边界模型，其中未知函数在一个域中求解方程，由于运动性现象，该域也是未知的。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学有关。参与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、生命和物理科学的交汇点上工作。首席调查员还将教授和指导本科生，重点是将数学应用于其他学科。自由边界问题，如Stefan问题、Hele-Shaw问题和Muscat问题，由于从数学的角度具有挑战性和重要的应用而受到极大的关注。本项目的重点是严格分析最近发展起来的活性凝胶(处于非平衡状态的凝胶)的自由边界偏微分方程(PDE)模型。与经典的自由边界问题和最新的肿瘤生长模型不同，这些模型是由非线性偏微分方程组控制的。主要研究人员将通过开发新的分析工具来分析自由边界下的分叉，从而严格分析定态、行波和旋转解的存在性和稳定性。这些溶液的稳定性对于生物物理应用是至关重要的，因为它允许人们区分在实验中很少观察到的稳定状态和不稳定状态。该项目将证明这些解的线性化稳定性分析通常是非决定性的，并将基于构造新的能量(Lyapunov型)泛函来开发真正的非线性稳定性分析的新技术。分析和数值结果将与对单个细胞和细胞聚集体的实验观察进行比较。一个长期目标是为推动重要生物学过程的细胞迁移提供理论上的理解，例如伤口愈合和癌症组织的侵袭。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.