UBC Theses and Dissertations
Analysis of pattern formation in reaction-diffusion models for cell polarization Liu, Yue
Small GTPases are a family of signalling proteins that regulates cell shape through actin assembly. Actin filaments are a major component of the cellular cytoskeleton, a supporting structure of the cell. Patterns of GTPase activity are related to a variety of cell behaviors, including cell polarization, formation of protrusions such as filopodia, and actin waves. These phenomena are in turn related to cell migration, cancer metastasis and viral infection. We seek to investigate pattern formation in the context of GTPase activity through a variant of the reaction-diffusion partial differential equation model, known as the Wave Pinning Model. By determining the possible spatio-temporal patterns, as well as their required conditions, we can better understand the cellular behaviors corresponding to these patterns. We first introduce and motivate the wave pinning model and its extensions involving F-actin feedback and source-sink terms. Next, we explore the behavior of the model with numerical simulations. We have identified patterns of localized spots, travelling waves, pulses, and others. We interpret some of these patterns as cell polarization, filopodia formation, and actin waves, as well as other complex behaviors that do not closely resemble common experimental observations. Next, we delineate distinct parameter regimes with Turing analysis and local perturbation analysis (LPA). Turing analysis is a classical method for determining linear stability of homogeneous steady states. LPA is a recently developed technique that is able to detect instabilities to perturbations of finite size that cannot be detected by Turing analysis. LPA has the caveat that it describes only the limiting behavior of the system as the ratio of diffusion coefficient of the two forms of GTPase goes to zero. We also study the effects stemming from the interaction of GTPase dynamics and the deforming cell boundary with a simplified wave pinning model coupled with cell size, and with Cellular Potts Model simulations coupling the full wave pinning model with the cell boundary. The results of the analysis allow us to comment on the impact of cell geometry, the various terms in our model and their interactions on pattern formation and cell behavior.
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