Shape Fluctuation Phase Transition of Lipid Membranes with Internal Degrees of Freedom in Aqueous Solution

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1.A first-order phase transition is found in the tethered surface model, which is defined by the Gaussian bond potential and the bending energy. The smooth phase of the model is separated from the crumpled phase by the transition on triangulated spherical surfaces. The first-order transition is independent of how the bending energy is discretized, and it is also independent of how the surface is triangulated.2.The above-mentioned first-order transition is independent of whether the bending energy is the extrinsic one or the intrinsic one.3.The above-mentioned first-order transition is independent of whether the surface is sphere or torus.4.The first-order transition of a tethered surface model changes to a second-order one when the surface is expanded with two point boundaries of fixed distance. The model is the one that is defined by the Gaussian bond potential and the intrinsic bending energy.