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The mechanical behavior of a closed membrane composed of two layers in contact is described as it is obtained by finding the minimum of the membrane bending energy at constant membrane area, constant difference between the areas of the two layers, and constant enclosed volume. It is shown that the membrane bending energy is a discontinuous function of the volume (v) and difference between the layer areas (delta a) defined relative to the volume and area difference of a sphere with the same membrane area, respectively. However, for different classes of shapes it is possible to obtain regions in the v/delta a diagram within which the shapes change continuously with v and delta a. These regions are shown for the egg, dumbbell and cup shape classes, respectively. The results of the shape analysis are used in the discussion of cell polarity, cytokinesis and gastrulation. Cell polarity is related to the decrease in symmetry during the transition from the radially symmetrical spherical shape to the asymmetrical shapes of the egg class. It is proposed that symmetrical cytokinesis occurs within the dumbbell class and that asymmetrical cytokinesis occurs within the egg class. Gastrulation is described as shape transformations within the class of cup shapes.

Mechanical behavior of closed lamellar membranes as a possible common mechanism for the establishment of developmental shapes.