The polarization and motility of eukaryotic cells depends on assembly and contraction of the actin cytoskeleton and its regulation by proteins called GTPases. The activity of GTPases causes assembly of filamentous actin (by GTPases Cdc42, Rac), resulting in protrusion of the cell edge. Mathematical models for GTPase dynamics address the spontaneous formation of patterns and nonuniform spatial distributions of such proteins in the cell. Here we revisit the wave-pinning model for GTPase-induced cell polarization, together with a number of extensions proposed in the literature. These include introduction of sources and sinks of active and inactive GTPase (by the group of A. Champneys), and negative feedback from F-actin to GTPase activity. We discuss these extensions singly and in combination, in 1D, and 2D static domains. We then show how the patterns that form (spots, waves, and spirals) interact with cell boundaries to create a variety of interesting and dynamic cell shapes and motion.

真核细胞的极化和运动取决于肌动蛋白细胞骨架的组装和收缩及其对称为GTPases的蛋白质的调节。GTPases的活性导致丝状肌动蛋白的组装（由GTPases Cdc42，Rac负责），导致细胞边缘突出。GTPase动力学的数学模型解决了细胞中此类蛋白质的自发形成模式和空间分布不均匀的问题。在这里，我们重新审视GTPase诱导的细胞极化的波固定模型，以及在文献中提出的许多扩展。其中包括引入有活性和无活性GTPase的来源和汇（由A. Champneys组），以及F-肌动蛋白对GTPase活性的负反馈。我们在1D和2D静态域中单独或组合讨论这些扩展。