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The Helfrich expression for the isotropic membrane bending energy was generalized for the case of anisotropic membranes by taking into account two intrinsic (spontaneous) curvatures, i.e., the intrinsic mean curvature H m and the intrinsic curvature deviator D m. Using this generalized expression for the membrane bending energy the shape equation for closed axisymmetric anisotropic membranes is solved numerically for the case of undulated tubular shapes. It is shown that the variation of one of the two intrinsic curvatures, H m or D m , may induce the first-order-like shape transitions between the undulated tubular membrane structures. The predicted discontinuous shape transitions were applied to explain the beading transitions without stretching, which were recently observed in nerve fibres.

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Amphiphile-induced tubular budding of the erythrocyte membrane was studied using transmission electron microscopy. No chiral patterns of the intramembraneous particles were found, either on the cylindrical buds, or on the tubular nanoexovesicles. In agreement with these observations, the tubular budding may be explained by in-plane ordering of anisotropic membrane inclusions in the buds where the difference between the principal membrane curvatures is very large. In contrast to previously reported theories, no direct external mechanical force is needed to explain tubular budding of the bilayer membrane.