Resumo: Lipid membranes, which are ubiquitous objects in biological environments are often confined. For example, they can be sandwiched between a substrate and the cytoskeleton between cell adhesion, or between other membranes in stacks, or in the Golgi apparatus. We present a study of the nonlinear dynamics of membranes in a model system, where the membrane is confined between two flat walls. The walls can be permeable or impermeable. We first derive a lubrication model for the membrane dynamics. The resulting dynamics is highly nonlinear and nonlocal. The solution of this model in one dimension exhibits frozen states due to oscillatory interactions between membranes caused by the bending rigidity. In two dimensions, the dynamics is more complex, and depends strongly on the amount of excess area in the system. Small area membranes form finite-size flat adhesion domains. In contrast, large area membrane forms of phase densely covered by wrinkles leading to a labyrinthine pattern and which do not evolve with time. In the intermediate membrane area regime, we find the coexistence of adhesion domains and wrinkles domains with coarsening. For small and intermediate membrane area, the effective membrane tension tends to two special values which can be related to the linear and nonlinear stability of flat fronts in the Swift-Hohenberg equation.
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