Phase-contrast microscopy is used to monitor the shapes of micron-scale fluid-phase phospholipid-bilayer vesicles in an aqueous solution. At fixed temperature, each vesicle undergoes thermal shape fluctuations. We are able, experimentally, to characterize the thermal shape ensemble by digitizing the vesicle outline in real time and storing the time sequence of Images. Analysis of this ensemble using the area-difference-elasticity (ADE) model of vesicle shapes allows us to associate (map) each time sequence to a point in the zero-temperature (shape) phase diagram. Changing the laboratory temperature modifies the control parameters (area, volume, etc.) of each vesicle, so it sweeps out a trajectory across the theoretical phase diagram. It is a nontrivial test of the ADE model to check that these trajectories remain confined to regions of the phase diagram where the corresponding shapes are locally stable. In particular, we study the thermal trajectories of three prolate vesicles which, upon heating, experienced a mechanical instability leading to budding. We verify that the position of the observed instability and the geometry of the budded shape are in reasonable accord with the theoretical predictions. The inability of previous experiments to detect the ''hidden'' control parameters (relaxed area difference and spontaneous cunvature) make this the first direct quantitative confrontation between vesicle-shape theory and experiment.
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