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Theory for the bending anisotropy of lipid membranes and tubule formation.

We study the spontaneous symmetry breaking of the bending rigidity of lipid membranes in two principal directions by a Landau mean-field theory. When the temperature is below the tilting temperature (T(c)), the coupling between molecular orientation and membrane local curvature square leads to an increase of the bending rigidity in the tilting direction and therefore a spontaneous symmetry breaking in two principal directions. The asymmetry (Delta) of the bending rigidity undergoes a continuous change upon cooling and grows as T(c)-T for T<T(c). We discuss this anisotropical effect on the tilt structure of the ripple phase P(beta(')) of a nearly flat membrane and the sphere-to-tubule transition in a dilute solution of lipids. The transition between the spherical vesicle phase and the tubular phase is predicted to be first order.