Stability and Bifurcations in Free-Boundary Models of Active Gels

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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.