Phase-contrast microscopy is used to monitor the shapes of micron-scalefluid-phase phospholipid-bilayer vesicles in aqueous solution. At fixedtemperature, each vesicle undergoes thermal shape fluctuations. We are ableexperimentally to characterize the thermal shape ensemble by digitizing thevesicle outline in real time and storing the time-sequence of images. Analysisof this ensemble using the area-difference-elasticity (ADE) model of vesicleshapes allows us to associate (map) each time-sequence to a point in thezero-temperature (shape) phase diagram. Changing the laboratory temperaturemodifies the control parameters (area, volume, etc.) of each vesicle, so itsweeps out a trajectory across the theoretical phase diagram. It is anontrivial test of the ADE model to check that these trajectories remainconfined to regions of the phase diagram where the corresponding shapes arelocally stable. In particular, we study the thermal trajectories of threeprolate vesicles which, upon heating, experienced a mechanical instabilityleading to budding. We verify that the position of the observed instability andthe geometry of the budded shape are in reasonable accord with the theoreticalpredictions. The inability of previous experiments to detect the ``hidden''control parameters (relaxed area difference and spontaneous curvature) makethis the first direct quantitative confrontation between vesicle-shape theoryand experiment.
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