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Abstract

Cell motility is a necessary process in many bodily functions, including wound healing. This motility is induced by a protein called actin, which forms filaments near the surface of cells. Regulators of filamentous actin formation include GTPases that act as molecular switches, cycling between active and inactive forms. Together, these proteins form complex spatio-temporal patterns (denoted “actin waves”) that govern cell motility. Recent experiments show that stationary patterns that induce directed cell motion and wave patterns that induce cell turning or ruffling can coexist in cells. This motivates the main question of this thesis: What are the underlying mechanisms that govern the coexistence of polar and wave patterns? To address this question, I derive a simple model for actin waves consisting of three partial differential equations for a mass-conserved active and inactive GTPase that promotes the formation of actin filaments (F-actin). The F-actin then feeds back onto the GTPase by increasing the inactivation rate. The simple model with its geometry (1D periodic cell perimeter) allows for the use of bifurcation analysis to study how patterns emerge, interact, coexist, and affect each other’s stability. This thesis focuses mostly on the long wavelength (LW) and finite-wavenumber Hopf (WB) bifurcations that lead to stationary and travelling wave solutions, respectively. I address the coexistence of these patterns by investigating the steady state solutions that emerge from two codimension-2 LW/WB instabilities. Only one codimension-2 instability leads to coexistence of polar and travelling wave solutions, and the analysis suggests that the initial bifurcating solutions need to form holes (i.e., troughs) instead of peaks. I also describe a rich structure of propagating highly localized patterns, namely, travelling fronts and excitable pulses. I show that mass conservation affects the types of patterns that emerge and discuss its effect on pattern formation. I supplement the bifurcation analysis with time-dependent simulations in one and two spatial dimensions, and simulations of motile cells to demonstrate the robustness and transient interactions between the coexisting states. The bifurcation analysis and time-dependent simulations show that the negative feedback from F-actin onto the GTPase affects the types of patterns that emerge.